Michael Froehner
email: michael.froehner@uniklinikum-dresden.de

Matt T Rosenberg

Does prostate cancer aggressiveness change with increasing age?

Keywords: prostate cancer, survival, age, biology, radical prostatectomy, active surveillance, external beam radiotherapy, randomized trials

Does prostate cancer aggressiveness change with

increasing age?

In the United States and many European countries, prostate cancer is the most common non-dermatological malignancy in males [START_REF] Jemal | Cancer Statistics[END_REF]. Old age is a well-established risk factor for prostate cancer. More than half of incident prostate cancer cases and more than 90 % of deaths from prostate cancer occur in men aged 65 years or older [START_REF] Altekruse | SEER Cancer Statistics Review, 1975-2007[END_REF]. Since the life expectancy continues to rise in most developed countries due to decreasing old-age mortality [START_REF] Christensen | Ageing populations: the challenges ahead[END_REF], men will be increasingly likely to live long enough to develop symptomatic or even fatal prostate cancer. Prostate cancer in the elderly therefore represents an increasingly important health problem. The relationship between age and prostate cancer biology is, however, still subject to controversies [START_REF] Sun | Men older than 70 years have higher risk prostate cancer and poorer survival in the early and late prostate specific antigen eras[END_REF][START_REF] Froehner | Age and prostate cancer survival[END_REF].

Data from randomized trials

In the European Randomized Study of Screening for Prostate Cancer (ERSPC), prostate-specific-antigen-(PSA)-based screening was found to reduce the mortality from prostate cancer for the entire study population by 20 % [START_REF] Schröder | Screening and prostate-cancer mortality in a randomized European study[END_REF]. This effect was, however, not demonstrable for study participants aged 70-74 years (the oldest age group included in this trial), among whom the rate ratio for death from prostate cancer in the screening arm was 1.26 (95 % confidence interval: 0.80-1.99) compared to no screening [START_REF] Schröder | Screening and prostate-cancer mortality in a randomized European study[END_REF]. These data suggests that men may not benefit from prostate cancer screening beyond their 70 th year of life.

In the only currently available randomized trial comparing radical prostatectomy with watchful waiting in men with clinically diagnosed prostate cancer, there were no detectable differences in disease-specific and overall survival rates between both treatment arms in the subset of men aged 65 years or older. Surprisingly, patients aged 65 years or older assigned to the watchful waiting arm were less likely to develop metastases or to die from prostate cancer, compared with their younger conservatively treated counterparts [START_REF] Bill-Axelson | Radical prostatectomy versus watchful waiting in localized prostate cancer: the Scandinavian prostate cancer group-4 randomized trial[END_REF]. Concerning disease-specific survival and metastatic progression, the survival curves in men aged 65 years or older whether of watchful waiting or surgery were very similar to that in younger ones assigned to radical prostatectomy.

Subgroup analysis showed that the benefit of the radical prostatectomy was limited to younger men and that watchful waiting resulted in worse outcomes (concerning metastatic progression and survival) in patients younger than 65 years (compared with the radical prostatectomy arm in this younger subgroup and to both treatment arms in men aged 65 or more years). The authors caution that the study was not designed to look at specific age groups and note that they have no data to judge biologic relevance of this age threshold. They further noted that the effect was not statistically significant. However, they did hypothesize that if the patients had reached an advanced age, potentially lethal tumors were more likely of already having metastasized and therefore being incurable [START_REF] Bill-Axelson | Radical prostatectomy versus watchful waiting in localized prostate cancer: the Scandinavian prostate cancer group-4 randomized trial[END_REF]. Compared with younger patients, in those aged 65 or more years undergoing watchful waiting, there was, however, a decreased rate of metastatic and eventually fatal progression [START_REF] Bill-Axelson | Radical prostatectomy versus watchful waiting in localized prostate cancer: the Scandinavian prostate cancer group-4 randomized trial[END_REF], calling this hypothesis into question.

Results of observational studies

In a series of 412 patients with metastatic prostate cancer, young age was an independent adverse predictor of disease progression and mortality [START_REF] Nayyar | Prognostic factors affecting progression and survival in metastatic prostate cancer[END_REF] suggesting a faster progression of metastatic disease in younger patients. If confirmed in different samples, this finding could be an alternative explanation of the restriction of the treatment effect to younger men in the cited Scandinavian randomized trial [START_REF] Bill-Axelson | Radical prostatectomy versus watchful waiting in localized prostate cancer: the Scandinavian prostate cancer group-4 randomized trial[END_REF]. There In a recent study, the outcome of metastatic prostate cancer was scarcely modified by age, whereas in locally advanced non-metastatic (cT4N0M0) disease there was a higher survival rate in younger men [START_REF] Hsiao | Stage IV Prostate Cancer: Survival Differences in Clinical T4, Nodal and Metastatic Disease[END_REF]. This finding led the authors to conclude that surgery offers better outcomes in this age group and should be offered as part of multimodal therapy. No theory was offered as to why a benefit was seen [START_REF] Hsiao | Stage IV Prostate Cancer: Survival Differences in Clinical T4, Nodal and Metastatic Disease[END_REF]. In another recently published very large study, higher age was associated with poorer prostate cancer-specific survival in univariate but not in multivariate analysis (after controlling for treatment modality and disease risk) [START_REF] Bechis | Impact of age at diagnosis on prostate cancer treatment and survival[END_REF].

In patients from the pre-PSA era, age was not associated with prostate cancerspecific survival neither after radical prostatectomy nor after external beam radiotherapy or with expectant management [START_REF] Barry | Outcomes for men with clinically nonmetastatic prostate carcinoma managed with radical prostatectomy, external beam radiotherapy, or expectant management: a retrospective analysis[END_REF]. Also in a recently published series of conservatively treated prostate cancer patients aged 65 years or older diagnosed between 1992 and 2002, the probability to succumb to their cancer within 10 years did narrowly not differ after subdivision into age groups [START_REF] Albertsen | Outcomes of localized prostate cancer following conservative management[END_REF]. Another study revealed a higher progression rate and a lower disease-specific survival among patients who underwent radical prostatectomy at an age of more than 70 years, compared with their younger counterparts in univariate but not in multivariate analysis (after controlling for the less favorable tumor-related risk profile in the elderly cohort) [START_REF] Froehner | Age and prostate cancer survival[END_REF].

Among high-risk patients (those with a Gleason score of 8-10 or positive lymph nodes), impaired disease-specific survival rates were observed in elderly men, whereas there were no such differences in the lower risk subgroups [START_REF] Froehner | Survival analysis in men undergoing radical prostatectomy at an age of 70 years or older[END_REF]. Similar observations were made by d'Amico and co-workers after external beam radiotherapy for clinically localized prostate cancer where older age was associated with a significantly worse prostate cancer-specific survival [START_REF] D'amico | Advanced age at diagnosis is an independent predictor of time to death from prostate carcinoma for patients undergoing external beam radiation therapy for clinically localized prostate carcinoma[END_REF]. To further add to the controversy, several studies demonstrated an adverse impact of higher age on prostate cancer progression rates after radical prostatectomy. After radical prostatectomy, a decreased progression rate in men younger than 40 years has been reported [START_REF] Loeb | Progression after radical prostatectomy for men in their thirties compared to older men[END_REF] as well as higher biochemical tumor control rates in men younger than 50 years [START_REF] Khan | Long-term cancer control of radical prostatectomy in men younger than 50 years of age: update 2003[END_REF], aged 60 years or younger [START_REF] Jayachandran | Risk stratification for biochemical recurrence in men with positive surgical margins or extracapsular disease after radical prostatectomy: results from the SEARCH database[END_REF][START_REF] Kane | Outcomes after radical prostatectomy among men who are candidates for active surveillance: Results from the SEARCH database[END_REF] and aged 70 years or younger [START_REF] Öbek | Age as a prognostic factor for disease recurrence after radical prostatectomy[END_REF], respectively. Richstone and co-workers observed a higher proportion of adverse tumor-related prognostic factors (Gleason score, clinical and pathological stages) in men aged 70 years or older compared with their younger counterparts but no differences in biochemical disease-free and cancer-specific survival rates [START_REF] Richstone | Radical prostatectomy in men aged >or=70 years: effect of age on upgrading, upstaging, and the accuracy of a preoperative nomogram[END_REF].

Less favorable tumor-related prognostic factors but no differences in systemic progression and prostate cancer-specific survival rates were made in another large study [START_REF] Siddiqui | Impact of patient age at treatment on outcome following radical retropubic prostatectomy for prostate cancer[END_REF]. In two series with narrowly 5.000 patients each tumors with adverse prognostic criteria were also more frequent in elderly men [START_REF] Froehner | Age and prostate cancer survival[END_REF][START_REF] Siddiqui | Impact of patient age at treatment on outcome following radical retropubic prostatectomy for prostate cancer[END_REF], whereas smaller studies did not reveal such differences [START_REF] Froehner | Survival analysis in men undergoing radical prostatectomy at an age of 70 years or older[END_REF][START_REF] Kane | Outcomes after radical prostatectomy among men who are candidates for active surveillance: Results from the SEARCH database[END_REF].

Explanations for the reported associations between age and prostate cancer outcome

From the available data, identifying a consistent pattern of interaction between age and prostate cancer aggressiveness is difficult. It is relatively well documented that there is an increasing proportion of adverse tumor-related prognostic factors with increasing age [START_REF] Froehner | Age and prostate cancer survival[END_REF][START_REF] Bechis | Impact of age at diagnosis on prostate cancer treatment and survival[END_REF][START_REF] Richstone | Radical prostatectomy in men aged >or=70 years: effect of age on upgrading, upstaging, and the accuracy of a preoperative nomogram[END_REF][START_REF] Siddiqui | Impact of patient age at treatment on outcome following radical retropubic prostatectomy for prostate cancer[END_REF][START_REF] Borek | Relationship of age to histologic grade in prostate cancer[END_REF] and a majority of studies seem to suggest an adverse impact of higher age on prostate cancer-related outcome [START_REF] Froehner | Age and prostate cancer survival[END_REF][START_REF] Hsiao | Stage IV Prostate Cancer: Survival Differences in Clinical T4, Nodal and Metastatic Disease[END_REF][START_REF] Bechis | Impact of age at diagnosis on prostate cancer treatment and survival[END_REF][START_REF] Froehner | Survival analysis in men undergoing radical prostatectomy at an age of 70 years or older[END_REF][START_REF] D'amico | Advanced age at diagnosis is an independent predictor of time to death from prostate carcinoma for patients undergoing external beam radiation therapy for clinically localized prostate carcinoma[END_REF][START_REF] Loeb | Progression after radical prostatectomy for men in their thirties compared to older men[END_REF][START_REF] Khan | Long-term cancer control of radical prostatectomy in men younger than 50 years of age: update 2003[END_REF][START_REF] Jayachandran | Risk stratification for biochemical recurrence in men with positive surgical margins or extracapsular disease after radical prostatectomy: results from the SEARCH database[END_REF][START_REF] Kane | Outcomes after radical prostatectomy among men who are candidates for active surveillance: Results from the SEARCH database[END_REF][START_REF] Öbek | Age as a prognostic factor for disease recurrence after radical prostatectomy[END_REF]. However, there is also some evidence for the opposite [START_REF] Bill-Axelson | Radical prostatectomy versus watchful waiting in localized prostate cancer: the Scandinavian prostate cancer group-4 randomized trial[END_REF][START_REF] Nayyar | Prognostic factors affecting progression and survival in metastatic prostate cancer[END_REF]. With the currently available data, it is not possible to explain the surprising lack of efficacy of radical prostatectomy versus watchful waiting in men aged 65 years or older in the Scandinavian randomized trial comparing radical prostatectomy with watchful waiting [START_REF] Bill-Axelson | Radical prostatectomy versus watchful waiting in localized prostate cancer: the Scandinavian prostate cancer group-4 randomized trial[END_REF]. The different distribution of risk factors for disease progression and mortality between older and younger patients in some studies [START_REF] Froehner | Age and prostate cancer survival[END_REF][START_REF] Richstone | Radical prostatectomy in men aged >or=70 years: effect of age on upgrading, upstaging, and the accuracy of a preoperative nomogram[END_REF][START_REF] Siddiqui | Impact of patient age at treatment on outcome following radical retropubic prostatectomy for prostate cancer[END_REF][START_REF] Borek | Relationship of age to histologic grade in prostate cancer[END_REF] age-related differences is yet unknown [START_REF] Siddiqui | Impact of patient age at treatment on outcome following radical retropubic prostatectomy for prostate cancer[END_REF].

Future studies

To obtain reliable answers to the question whether increasing age modifies the aggressiveness of early prostate cancer, more information from randomized trials would be desirable. Appropriately designed studies could investigate such issues as whether tumor progression rates are actually lower in elderly conservatively treated patients as the Scandinavian randomized trial [START_REF] Bill-Axelson | Radical prostatectomy versus watchful waiting in localized prostate cancer: the Scandinavian prostate cancer group-4 randomized trial[END_REF]

  , also data showing poorer survival in elderly men with prostate cancer.

  suggested or whether there are subgroups who are at a high risk of progression and could benefit from immediate patients with an adverse tumor-related risk profile. Despite randomization, such kind of selection might result in an enrichment of young low risk cases and simultaneously elderly high risk cases making comparisons between age groups difficult to interpret.ConclusionAt present, identifying a consistent pattern of interaction between age and prostate cancer aggressiveness is difficult. Selection may have confounded particularly nonrandomized studies. Randomized trials comparing competing treatment options in elderly men with early prostate cancer that are adequately powered to enable meaningful subgroup analyses to investigate the relationship between age and prostate cancer biology.

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curative treatment also at an advanced age. Even randomized trials, however, might be affected by a selection bias caused by an earlier detection of cancers by screening in younger men and a preference of aggressive treatment in elderly

Key messages

With the currently available data, it is difficult to identify a consistent pattern of interaction between age and prostate cancer aggressiveness.

Altogether, a majority of studies seem to suggest an adverse impact of higher age on prostate cancer-related outcome, there is, however, also some evidence for the opposite. Currently, it is not possible to explain the disturbing lack of efficacy of radical prostatectomy versus watchful waiting in men aged 65 years or older in the Scandinavian randomized trial comparing radical prostatectomy with watchful waiting sufficiently.


MD Hans-Christoph Diener
email: hans.diener@uni-duisburg-essen.de

Diener Hc

Weimar

Secondary prevention in the acute and early chronic phase after ischemic stroke and transient ischemic attacks with antiplatelet drugs -is antiplatelet monotherapy still reasonable?

Keywords: transient ischemic attack, ischemic stroke, secondary prevention, aspirin, clopidogrel, dipyridamole

Secondary prevention in the acute and early chronic phase after ischemic stroke and transient ischemic attacks with antiplatelet drugs -is antiplatelet monotherapy still reasonable?

Introduction

Intravenous thrombolysis with recombinant tissue plasminogen activator is the only proven medical treatment for acute ischemic stroke in a time period up to 4.5 hours after onset of stroke symptoms. However, only a minority of patients is eligible for intravenous thrombolysis due to delayed presentation or contraindications to thrombolysis. In the remaining patients, early prevention of recurrent stroke is one of the primary therapeutic targets. Numerous prospective studies have shown that the risk of a subsequent or recurrent ischemic stroke is highest in the first 48 to 72 hours after the initial cerebrovascular ischemic event [START_REF] Giles | Risk of stroke early after transient ischaemic attack: a systematic review and meta-analysis[END_REF].

We discuss current evidence from randomised trials concerning secondary stroke prevention with antiplatelets in the acute and early chronic phase after TIA and stroke.

Secondary stroke prevention with aspirin in the acute and early chronic postischemic phase

Early treatment with aspirin in patients with acute ischemic stroke was investigated in two large clinical trials, the International Stroke Trial (IST) and the Chinese Acute Stroke Trial (CAST). IST was a randomised, unblinded trial that recruited a total of 19435 patients with acute ischemic stroke to receive aspirin (300 mg/daily), one of two doses of subcutaneously administered heparin (5000 U/BID or 12500 U/BID), or both, in a factorial design within 48 hours of stroke symptom onset. Among aspirinallocated patients there were non-significantly fewer deaths within 14 days (9.0% vs. with a non-significant increase in hemorrhagic strokes (1.1% vs. 0.9).

In a prespecified combined analysis of IST and CAST, aspirin treatment started in the acute post-ischemic stroke phase resulted in a reduction of 10 deaths or recurrent strokes and causes 2 hemorrhagic strokes per 1000 treated patients [START_REF] Chen | Indications for early aspirin use in acute ischemic stroke. A combined analysis of 40 000 randomized patients from the Chinese acute stroke trial and the International stroke trial[END_REF].

Abciximab in early secondary prevention

The intravenous use of the platelet glycoprotein IIb/IIIa inhibitor abciximab has been considered to be safe when administered within 24 hours after ischemic stroke onset after two phase II trials had been carried out. However, the phase III trial AbESTT-II had to be terminated prematurely after enrolment of 808 patients due to a significantly increased bleeding rate in the active treatment group [START_REF] Adams | Emergency administration of abciximab for treatment of patients with acute ischemic stroke: results of an international phase III trial: Abciximab in Emergency Treatment of Stroke Trial (AbESTT-II)[END_REF]. During the first 5 days of enrolment, 5.5% of patients who had received intravenously administered abciximab within 5 hours of onset of stroke had symptomatic or fatal intracranial hemorrhage versus 0.5% of placebo-treated patients. The combination of aspirin plus dipyridamole has been shown to be more effective than aspirin monotherapy in long-term secondary stroke prevention in the randomised trials ESPS-2 and ESPRIT [START_REF] Diener | European Stroke Prevention Study 2. Dipyridamole and acetylsalicylic acid in the secondary prevention of stroke[END_REF][START_REF] The | Aspirin plus dipyridamole versus aspirin alone after cerebral ischaemia of arterial origin (ESPRIT): randomised controlled trial[END_REF]. However, enrolment of patients in these trials occurred up to 6 months after the cerebrovascular ischemic event. The combination of aspirin (25 mg BID) and extended-release dipyridamole (200 mg BID) was compared with clopidogrel monotherapy (75 mg daily) in more than 20000 patients in the randomised PRoFESS trial [START_REF] Sacco | Randomized comparison of aspirin and extended-release dipyridamole to copidogrel for prevention of recurrent strokes[END_REF]. A post-hoc analysis of this trial investigated the 1360 patients who were randomised within 72 hours of ischemic stroke onset [START_REF] Bath | Effect of combined aspirin and extended-release dipyridamole versus clopidogrel on functional outcome and recurrence in acute, mild ischemic stroke: PRoFESS subgroup analysis[END_REF]. There was no significant difference in death or dependency at 30 days between both antiplatelet regimen. A non-significant trend towards a reduction of recurrent stroke was seen with the combination of aspirin and extended-release dipyridamole at 90 days (1.64% vs. 2.91%; OR 0.56, 95%CI 0.26 to 1.18).

The randomised EARLY trial compared the combination of aspirin (25 mg BID) and extended-release dipyridamole (200 mg BID) with aspirin monotherapy (100 mg/daily) started within 24 hours of symptom onset in patients with ischemic stroke (NIHSS score ≤ 20) or TIA [START_REF] Dengler | Early treatment with aspirin plus extended-release dipyridamole for transient ischaemic attack or ischaemic stroke within 24 h of symptom onset (EARLY trial): a randomised, open-label, blinded-endpoint trial[END_REF]. A total of 543 patients were randomised, 283 receiving aspirin and extended-release dipyridamole (early initiation group) and 260 receiving aspirin monotherapy for the initial 7 days (late initiation group). Thereafter, all patients were treated with the combination antiplatelet therapy. The primary endpoint was an excellent or good functional outcome at 90 days, defined as a value of 0 or 1 on the modified Rankin scale, which was assessed by a blinded investigator using a standardised telephone interview. Vascular events and mortality were also assessed as a composite safety and efficacy endpoint. At day 90, 56% of patients in the early initiation group and 52% of patients treated with aspirin monotherapy during the first 7 days had a modified Rankin scale value of 0 or 1, a difference that was not statisti-

Dual antiplatelet therapy with Clopidogrel and Aspirin in the acute and early chronic secondary stroke prevention

The FASTER trial was a randomised controlled pilot study that compared antiplatelet therapy with clopidogrel and aspirin with aspirin alone in TIA and stroke patients, within 24 hours of symptom onset [START_REF] Kennedy | Fast assessment of stroke and transient ischaemic attack to prevent early recurrence (FASTER): a randomised controlled pilot trial[END_REF]. 392 patients were randomised to an initial loading dose of 300 mg Clopidogrel followed by 75 mg daily or placebo. All patients received 180 mg aspirin daily. The primary outcome was an ischemic or hemorrhagic stroke within 90 days, which did not significantly differ between both treatment arms (7.1% on clopidogrel had an ischemic stroke vs. 10.8% on placebo). Intracranial hemorrhage occurred in 2 patients in the clopidogrel group. A combined analysis of patients treated within 24 hours in the FASTER, CHARISMA, CARESS and MATCH trials showed a 34% relative risk reduction in patients on clopidogrel and aspirin (figure 2). In the dual platelet therapy group, 29 out of 214 (13.5%) patients suffered a stroke versus 45 out of 213 (14.4%) patients in the aspirin monotherapy group [START_REF] Kennedy | Fast assessment of stroke and transient ischaemic attack to prevent early recurrence (FASTER): a randomised controlled pilot trial[END_REF].

A meta-analysis of the EARLY and the FASTER trials studied the combined outcome of death, myocardial infarction, ischemic stroke, TIA and intracranial hemorrhage. 57 of 481 (11.9%) patients on the combination therapy with aspirin plus dipyridamole or aspirin plus clopidogrel reached the endpoint, as compared to 80 of 454 (17.6%) on aspirin monotherapy (relative risk reduction 42%, p=0.014) (8) (figure 3).

Whereas the FASTER trial was a pilot study, the projected POINT study will recruit 4150 patients treated with a clopidogrel loading dose of 600 mg followed by daily 75 mg in combination with 50 -325 mg aspirin daily in comparison to aspirin monotherapy in patients with TIA and minor stroke initiated within 12 hours of symptom onset (http://clinicaltrials.gov/ct2/show/NCT00991029).

Bleeding risk in patients treated with a combination of clopidogrel and aspirin and potential rebound increase in risk of ischemic stroke recurrence

According to the MATCH study [START_REF] Diener | Acetylsalicylic acid on a background of clopidogrel in high-risk patients randomised after recent ischaemic stroke or transient ischaemic attack: The MATCH trial results[END_REF], combination of aspirin plus clopidogrel started in the acute phase should not be given over long term because the combination has no significant benefit but an increased risk of intracranial and systemic bleeding compli- FASTER studies [START_REF] Geraghty | Preliminary evidence of a high risk of bleeding on aspirin plus clopidogrel in aspirin-naive patients in the acute phase after TIA or minor ischaemic stroke[END_REF]. Major or life-threatening bleeding when treated with aspirin plus clopidogrel occurred in 9/241 aspirin-naive patients in 90 days (90-day risk = 4.8%, 95%CI 1.6-8.0) versus in 1/204 prior-aspirin patients (p = 0.009). Therefore, the early bleeding risk under aspirin plus clopidogrel seems to be a cause for concern mainly in aspirin-naive patients. These results were based on observational data and an unplanned subgroup analysis from the randomised FASTER trial. It therefore cannot be excluded that there are other differences between the patient groups unrelated to antiplatelet regimens.

In acute coronary syndrome, early discontinuation of clopidogrel results in a transient rebound increase in risk of recurrence [START_REF] Ho | Incidence of death and acute myocardial infarction associated with stopping clopidogrel after acute coronary syndrome[END_REF]. Geraghty et al. investigated whether there is a similar rebound effect in patients with TIA or stroke. In the EXPRESS study, a total of 320 patients were prescribed a 30-day course of aspirin and clopidogrel (both 75 mg daily) acutely after TIA or minor stroke [START_REF] Geraghty | Low risk of rebound events after a short course of clopidogrel in acute TIA or minor stroke[END_REF]. Clopidogrel was stopped after 30 days and aspirin was continued thereafter. There were 5 recurrent ischemic strokes and 7 TIAs during the aspirin and clopidogrel treatment period, but no strokes and 4 TIAs during the 30 days after stopping clopidogrel. A similar temporal trend in stroke risk was seen in the 487 patients prescribed aspirin alone in the acute phase, with 12 and 5 strokes in the equivalent time periods. The upper 95% CI of the observed 0% risk of stroke during the 30 days after stopping clopidogrel was 1.15%.

This finding suggests that there is unlikely to be a large rebound effect after discontinuation of a 30-day course of clopidogrel and aspirin in acute TIA and minor ischemic stroke. Considering the available evidence, patients with an acute TIA or minor ischemic stroke might benefit from combination of aspirin plus clopidogrel or aspirin plus dipyridamole for at least 30 days. However, one has to keep in mind that most of the available data has been derived from meta-analyses of non-prespecified subgroups from randomised trials. Thus, a possible bias or confounding factors cannot be excluded. There is good evidence from randomised trials for the combination of aspirin and dipyridamole both in the acute, early and long term secondary stroke prevention.

As shown before, a combined antiplatelet therapy with aspirin and clopidogrel for a limited time period might result in a lower recurrent stroke rate in the acute and early chronic post-ischemic stroke phase. However, this combination therapy might result in a higher bleeding rate in patients who were not treated with aspirin before. Thus, additional randomised trials are required for this combination and to identify patient subgroups that benefit most. In contrast, the combination of aspirin and clopidogrel has no benefit in long-term secondary stroke prevention but increases the bleeding risk.

Deleted: a combined therapy

  Secondary prevention in the acute and early chronic phase after ischemic stroke and transient ischemic attacks with antiplatelet drugs -is antiplatelet monotherapy still reasonable?

  cations. An important question is whether the combination of clopidogrel and aspirin increases the risk of intracranial bleeding during the acute post-ischemic phase as it does during long-term treatment. In order to estimate the early bleeding risk during the acute phase, Geraghty et al. investigated pooled data from the EXPRESS and Deleted:

  Figure legend

Figure 1 :

 1 Figure 1: Time to first event that was included in the composite endpoint of non-fatal stroke, TIA, non-fatal myocardial infarction and major bleeding in the EARLY trial. ASA = acetylsalicylic acid. ER-DP = extended released dipyridamole. HR = hazard ratio. Adopted from The Lancet Neurol, 9,159-166, Dengler R, Diener HC, Schwartz A, Grond M, Schumacher H, Machnig T, Eschenfelder CC, Leonard J, Weissenborn K, Kastrup A, Haberl R; EARLY Investigators. Early treatment with aspirin plus extended-release dipyridamole for transient ischaemic attack or ischaemic stroke within 24 h of symptom onset (EARLY trial): a randomised, open-label, blinded-endpoint trial. Copyright (2010), with permission from Elsevier.

Figure 2 :

 2 Figure 2: Fixed-effects meta-analysis of 90-day risk of the combined outcome of stroke, TIA, acute coronary syndrome, and all-cause death in stroke and TIA patients enrolled within 24 h of onset in the FASTER, CHARISMA, CARESS and MATCH trial. Note that x-axis is a logarithmic scale. Reprinted from The Lancet Neurol, 6, 961-969, Kennedy J, Hill MD, Ryckborst KJ, Eliasziw M, Demchuk AM, Buchan AM. Fast assessment of stroke and transient ischaemic attack to prevent early recurrence (FASTER): a randomised controlled pilot trial. Copyright (2007), with permission from Elsevier.

Figure 3 :

 3 Figure 3: Comparison of the composite endpoint non-fatal stroke, TIA, non-fatal myocardial infarction and major bleeding in the first 7 days in the EARLY and FASTER

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: Time to first event that was included in the composite endpoint of non-fatal stroke, TIA, non-fatal myocardial infarction and major bleeding in the EARLY trial. ASA = acetylsalicylic acid. ER-DP = extended released dipyridamole. HR = hazard ratio. Adopted from The Lancet Neurol, 9,159-166, Dengler R, Diener HC, Schwartz A, Grond M, Schumacher H, Machnig T, Eschenfelder CC, Leonard J, Weissenborn K, Kastrup A, Haberl R; EARLY Investigators. Early treatment with aspirin plus extend-ed-release dipyridamole for transient ischaemic attack or ischaemic stroke within 24 h of symptom onset (EAR-LY trial): a randomised, open-label, blinded-endpoint trial. Copyright (2010), with permission from Elsevier.189x129mm (300 x 300 DPI)

The combination of aspirin and extended-release dipyridamole in early secon- dary stroke prevention

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Conflict of interest


Sandi Klavžar
email: sandi.klavzar@fmf.uni-lj.si

Michel Mollard
email: michel.mollard@ujf-grenoble.fr

Wiener Index and Hosoya Polynomial of Fibonacci and Lucas Cubes

In the language of mathematical chemistry, Fibonacci cubes can be defined as the resonance graphs of fibonacenes. Lucas cubes form a symmetrization of Fibonacci cubes and appear as resonance graphs of cyclic polyphenantrenes. In this paper it is proved that the Wiener index of Fibonacci cubes can be written as the sum of products of four Fibonacci numbers which in turn yields a closed formula for the Wiener index of Fibonacci cubes. Asymptotic behavior of the average distance of Fibonacci cubes is obtained. The generating function of the sequence of ordered Hosoya polynomials of Fibonacci cubes is also deduced. Along the way, parallel results for Lucas cubes are given.

Introduction

In this paper we are interested in the Wiener index and more general distance properties of Fibonacci and Lucas cubes. Fibonacci cubes were introduced as interconnection networks [START_REF] Hsu | Fibonacci cubes-a new interconnection technology[END_REF] and later studied from many aspects, see the recent survey [START_REF] Klavžar | Structure of Fibonacci cubes: a survey[END_REF]. From our point of view it is important that Fibonacci cubes also play a role in mathematical chemistry: Fibonacci cubes are precisely the resonance graphs of fibonacenes which in turn form an important class of hexagonal chains [START_REF] Klavžar | Fibonacci cubes are the resonance graphs of Fibonaccenes[END_REF]. (For related results about resonance graphs, known also as Z-transformation graphs, see [START_REF] Ou | Determining which Fibonacci (p,r)-cubes can be Z-transformation graphs[END_REF][START_REF] Vesel | Characterization of resonance graphs of catacondensed hexagonal graphs[END_REF][START_REF] Zhang | Resonance graphs and a binary coding for the 1-factors of benzenoid systems[END_REF][START_REF] Zhang | Fibonacci-like cubes as Z-transformation graphs[END_REF].) Moreover, Lucas cubes also found chemical applications in [START_REF] Žigert | Lucas cubes and resonance graphs of cyclic polyphenantrenes[END_REF].

The Wiener index of a graph is the first and (one of) the most studied invariant(s) in mathematical chemistry, see for instance extensive surveys [START_REF] Dobrynin | Wiener index of trees: theory and applications[END_REF][START_REF] Dobrynin | Wiener index of hexagonal systems[END_REF] and recent papers [START_REF] Ashrafi | Computing Wiener and detour indices of a new type of nanostar dendrimers[END_REF][START_REF] Balakrishnan | A sharp lower bound for the Wiener index of a graph[END_REF][START_REF] Cohen | On Wiener index of graphs and their line graphs[END_REF][START_REF] Dobrynin | On the Wiener index of fibonacenes[END_REF][START_REF] Wu | Wiener index of line graphs[END_REF][START_REF] Wu | On the Wiener index of graphs[END_REF]. An equivalent approach to the Wiener index is to investigate the average distance of a graph, which is indeed frequently done in pure mathematics, cf. [START_REF] Dankelmann | Average distance and vertex-connectivity[END_REF][START_REF] Hinz | The average distance on the Sierpiński gasket[END_REF]. ¿From the recent papers on the Wiener index we point out that [START_REF] Dobrynin | On the Wiener index of fibonacenes[END_REF] contains many new results on the Wiener index of fibonacenes, the class of graphs that made Fibonacci cubes appealing in chemistry! Hence an investigation of the Wiener index and related invariants of Fibonacci cubes seems well justified.

We wish to add that results similar to ours were very recently obtained by Došlić in [START_REF] Došlić | Counting conjugated circuits in benzenoid chains[END_REF] while studying conjugated circuits in benzenoid chains. In particular, Fibonacci numbers convolved with themselves play a role here and there, and an average Kekulé structure of an n-ring fibonacene contains approximately 2nϕ(2ϕ + 1)/5 conjugated hexagons, where ϕ = 1+ √ 5 2 is the golden ration.

The paper is organized as follows. In the next section concepts and results needed are given.

Then, in Section 3, we prove that the Wiener index of Fibonacci cubes can be written as the sum of products of four Fibonacci numbers and use this result to deduce a closed formula for the Wiener index of Fibonacci cubes. Hence the Wiener index of Fibonacci cubes can be obtained in constant time which improves a result from [START_REF] Udaya Kumar | Computing Wiener index of Fibonacci cubes[END_REF] that computes it in O(log F n ) time. In addition, we also determine the asymptotic behavior of the average distance of Fibonacci cubes and obtain parallel results for Lucas cubes. In Section 4 we consider the Hosoya polynomial and obtain the generating function of the sequence of ordered Hosoya polynomials of Fibonacci cubes as well as the corresponding generating function for Lucas cubes. To make the paper of a reasonable length, we only indicate some of numerous possible uses of these generating functions.

The distance in this paper is the usual shortest-path distance-the number of edges on a shortest path between vertices. The Wiener index W (G) of a connected graph G is the sum of distances over all unordered pairs of vertices of G. The vertex set of the d-cube Q d , also called a hypercube of dimension d, is the set of all binary strings of length d, and two vertices are adjacent if they differ in precisely one position.

A Fibonacci string of length n is a binary string b 1 b 2 . . . b n with b i • b i+1 = 0 for 1 ≤ i < n.
The Fibonacci cube Γ n (n ≥ 1) is the subgraph of Q n induced by the Fibonacci strings of length n. For convenience we also consider the empty string and set

Γ 0 = K 1 . Call a Fibonacci string b 1 b 2 . . . b n a Lucas string if b 1 • b n = 0. Then the Lucas cube Λ n (n ≥ 1) is the subgraph of Q n
induced by the Lucas strings of length n. We also set Λ 0 = K 1 .

Let {F n } be the Fibonacci numbers:

F 0 = 0, F 1 = 1, F n = F n-1 + F n-2 for n ≥ 2. Let
F n be the Fibonacci strings of length n. Let F 0 n and F 1 n be the strings of F n that end with, respectively, 0 and 1.

A subgraph G of a graph H is an isometric subgraph if the distance between any vertices of G equals the distance between the same vertices in H. Isometric subgraphs of hypercubes are partial cubes. The dimension of a partial cube G is the smallest integer d such that G is an isometric subgraph of Q d . Many (chemically) important classes of graphs are partial cubes, in particular trees, median graphs, benzenoid graphs, phenylenes, grid graphs and bipartite torus graphs. In addition, Fibonacci and Lucas cubes are partial cubes as well, see [START_REF] Klavžar | On median nature and enumerative properties of Fibonacci-like cubes[END_REF].

Let a partial cube G of dimension k be given together with its isometric embedding into Q k .

Then for i = 1, 2, . . . , k and χ = 0, 1, the semicube W (i,χ) is defined as follows:

W (i,χ) (G) = {u = u 1 u 2 . . . u k ∈ V (G) | u i = χ} .
For a fixed i, the pair W (i,0) (G), W (i,1) (G) of semicubes is called a complementary pair of semicubes. It should be pointed out that since the embedding of G into Q k is unique (modulo permutation of coordinates), see [START_REF] Winkler | Isometric embedding in products of complete graphs[END_REF], the definition of the semicubes does not depend on the embedding. The following result from [START_REF] Klavžar | Labeling of benzenoid systems which reflects the vertexdistance relations[END_REF] that computes the Wiener index of a partial cube from the orders of their complementary pairs of semicubes will be our starting point in Section 3:

Theorem 2.1 Let G be a partial cube of dimension k isometrically embedded into Q k . Then W (G) = k i=1 |W (i,0) (G)| • |W (i,1) (G)| .
The Hosoya polynomial [START_REF] Ali | Hosoya polynomials of pentachains[END_REF][START_REF] Xu | Hosoya polynomials of TUC 4 C 8 (S) nanotubes[END_REF] (also known as the Wiener polynomial) H(G, x) of a connected graph G is the distance counting polynomial:

H(G, x) = {u,v} x d(u,v) .
We refer to [START_REF] Behmaran | Some new results on distance-based polynomials[END_REF] for several polynomials related to the Hosoya polynomial: edge-Wiener polynomial, Schultz polynomial, and Gutman polynomial. Sometimes it is more convenient to consider ordered Hosoya polynomial [START_REF] Mohamadinezhad-Rashti | Some new results on the Hosoya polynomial of graph operations[END_REF], that is, the counting polynomial of the distances among ordered pairs of vertices:

H(G, x) = (u,v)∈V (G)×V (G) x d(u,v) .
For example, when using Cartesian product, ordered polynomials are more adapted because we have the simple relation

H(G G , x) = H(G, x) • H(G , x).
The two polynomials are easily linked as follows:

2H(G, x) = H(G, x) -|V (G)| = H(G, x) -H(G, 0) .

Wiener index

We begin by expressing the Wiener index of Fibonacci cubes with Fibonacci numbers:

Theorem 3.1 For any n ≥ 0, W (Γ n ) = n i=1 F i F i+1 F n-i+1 F n-i+2 .
Proof. The result holds for n = 0 because W (Γ 0 ) = 0. Let n ≥ 1 and recall that Γ n is a partial cube. Moreover, the dimension of Γ n is n, hence by Theorem 2.1 we have

W (Γ n ) = n i=1 |W (i,1) (Γ n )| • |W (i,0) (Γ n )| . Consider the set W (i,1) (Γ n ), where 2 ≤ i ≤ n -1. Let b = b 1 . . . b n ∈ W (i,1) (Γ n ), then b i = 1 which implies that b i-1 = 0 and b i+1 = 0. It follows that for 2 ≤ i ≤ n -1, |W (i,1) (Γ n )| = |F i-2 | • |F n-i-1 |. Furthermore, |W (1,1) (Γ n )| = |W (n,1) (Γ n )| = |F n-2 |. Similarly, |W (i,0) (Γ n )| = |F i-1 | • |F n-i | for 2 ≤ i ≤ n -1 and |W (1,0) (Γ n )| = |W (n,0) (Γ n )| = |F n-1 |. Therefore, W (Γ n ) = n-1 i=2 (|F i-2 | • |F n-i-1 | • |F i-1 | • |F n-i |) + 2|F n-2 | • |F n-1 | = n i=1 F i F n-i+1 F i+1 F n-i+2 ,
where we have used that for any j,

|F j | = F j+2 . Note that the sequence {W (Γ n )} ∞ n=0 is the convolution of {F i F i+1 } ∞ i=0 with {F i+1 F i+2 } ∞ i=0 .
In order to obtain a closed formula for W (Γ n ) we will apply the appealing theory of Greene and Wilf presented in [START_REF] Greene | Closed form summation of C-finite sequences[END_REF]. They studied the existence of closed formulas for expressions of the form

n-1 j=0 G 1 (a 1 n + b 1 j + c 1 )G 2 (a 2 n + b 2 j + c 2 ) • • • G k (a k n + b k j + c k ) ,
where each sequence {G i (n)} is a sequence that satisfies a linear recurrence. Fibonacci numbers are of course such a sequence. Then it follows from [12, Theorem 2] that the sum

n-1 i=0 F i F n-i+1 F i+1 F n-i+2 can be expressed as a linear combination of F 2 n , F 2 n+1 , F n F n+1 , nF 2 n , nF 2 n+1
, and nF n F n+1 . Furthermore, [START_REF] Greene | Closed form summation of C-finite sequences[END_REF]Theorem 3] implies that if a linear combination of the six monomials and the sum agree for n = 0, 1, . . . , 5, then they agree for all n. Since the

values of n-1 i=0 F i F n-i+1 F i+1 F n-i+2
for n = 0, 1, . . . , 5 are 0, 0, 2, 10, 39, 136, the values of the linear combination a F

2 n + b F 2 n+1 + c F n F n+1 + d nF 2 n + e nF 2 n+1 + f nF n F n+1 must be         0 1 0 0 0 0 1 1 1 1 1
                a b c d e f         =         0 0 2 10 39 136         whose solution is {a, b, c, d, e, f } = { 4 25 , 0, - 23 25 , 4 25 , 6 25 , 9 25
} .

Hence

n-1 i=0 F i F n-i+1 F i+1 F n-i+2 = 4 25 F 2 n + 4 25 nF 2 n -23 25 F n F n+1 + 9 25 nF n F n+1 + 6 25 nF 2 n+1 . Note that W (Γ n ) = n i=1 F i F n-i+1 F i+1 F n-i+2 = F n F n+1 + n-1 i=0 F i F n-i+1 F i+1 F n-i+2 therefore: Theorem 3.2 For any n ≥ 0, W (Γ n ) = 4(n + 1)F 2 n 25 + (9n + 2)F n F n+1 25 + 6nF 2 n+1 25 .
The first values of the sequence {W (Γ n )} are 0, 1, 4, 16, 54, 176, 548, . . .

For a (connected) graph G, its average distance µ(G) is defined as

µ(G) = 1 |V (G)| 2 W (G) .
Theorem 3.2 yields the asymptotic behavior of the average distance of Fibonacci cubes:

Corollary 3.3 lim n→∞ µ(Γ n ) n = 2 5 .
Proof. From Binet's formula for the Fibonacci numbers it follows that

lim n→∞ F n+1 F n = ϕ = 1 + √ 5 2 .
Then Theorem 3.2 and the fact that

|V (Γ n )| = F n+2 imply: lim n→∞ µ(Γ n ) n = 2 4 25ϕ 4 + 9 25ϕ 3 + 6 25ϕ 2 = 2 5
.

Obtaining the Wiener index of Lucas cubes is significantly simpler than the Wiener index of Fibonacci cubes. The intrinsic reason for it is that in Lucas cubes all the coordinates of vertices are equivalent, while in Fibonacci cubes the first and the last coordinate behave differently from the other coordinates.

Theorem 3.4 For any

n ≥ 1, W (Λ n ) = n F n-1 F n+1 .
Proof. We again use the fact that Lucas cubes are partial cubes [START_REF] Klavžar | On median nature and enumerative properties of Fibonacci-like cubes[END_REF] and hence Theorem 2.1 can be applied, that is,

W (Λ n ) = n i=1 |W (i,1) (Λ n )| • |W (i,0) (Λ n )| .
Considering Lucas strings on a circle we infer that

|W (i,1) | • |W (i,

Hosoya polynomial

In this section we make a more general approach and consider the ordered Hosoya polynomial of Fibonacci cubes H(Γ n , x) and its generating function

f (x, z) = n≥0 H(Γ n , x)z n = n,k≥0 f n,k x k z n , where f n,k is the number of pairs of vertices (u, v) ∈ V (Γ n ) × V (Γ n ) such that d(u, v) = k.
Theorem 4.1 The generating function of the sequence of ordered Hosoya polynomials of Γ n is

f (x, z) = n≥0 H(Γ n , x)z n = 1 + z + xz(1 -z) 1 -z -z 2 + xz(-1 -z + z 2 ) . Proof. For i, j ∈ {0, 1}, let f ij n,k be the number of (u, v) ∈ V (Γ n ) × V (Γ n ) such that d(u, v)
= k where u ends with i and v ends with j. Let us consider the generating polynomials

f ij (x, z) = n,k≥0 f ij n,k x k z n .
Then, having in mind the empty string,

f (x, z) = 1 + f 00 (x, z) + f 01 (x, z) + f 10 (x, z) + f 11 (x, z) . ( 1
)
By symmetry,

f 01 (x, z) = f 10 (x, z) . ( 2
)
If u and v end with 1 then they are the concatenation of an arbitrary Fibonacci string of F n-2

with 01 thus f 11 n,k = f n-2,k for n ≥ 2. Using the initial conditions f 11 1,0 = 1 and f 11 1,1 = f 11 0,0 = 0 we obtain by a standard computation that

f 11 (x, z) = z 2 f (x, z) + z . ( 3
)
If u and v end with 0 then they are both the concatenation of an arbitrary Fibonacci string of

F n-1 with 0 thus f 00 n,k = f n-1,k for n ≥ 1.
Using the initial condition f 00 0,0 = 0 we obtain

f 00 (x, z) = zf (x, z) . ( 4
)
For n ≥ 2, if u ends with 1 and v ends with 0, then u is the concatenation of a Fibonacci string of F 0 n-1 with 1, and v is the concatenation of a string of

F 0 n-1 ∪F 1 n-1 with 0. Therefore f 10 n,k = f 00 n-1,k-1 + f 01 n-1,k-1 .
Using the initial conditions f 10 1,1 = 1 and f 10 1,0 = f 10 0,0 = 0 we obtain by summation that

f 10 (x, z) -xz = xzf 00 (x, z) + xzf 10 (x, z)
and thus

f 10 (x, z) = xzf 00 (x, z) + xz 1 -xz = xz 2 f (x, z) + xz 1 -xz . ( 5
)
Using the symmetry (2), by substitution of the expressions (3), ( 4) and ( 5) in ( 1) we obtain the relation

f (x, z)(1 -z - 2xz 2 1 -xz -z 2 ) = 1 + z + 2xz 1 -xz thus the theorem.
Theorem 4.1 can be applied in many different ways, let us mention some of them. By

definition of H(G, x) we have that H(G, 0) = |V (G)|. Furthermore, it is well-know that H (G, 1) = 2W (G), cf. [5, 13]. Theorem 4.1 gives that f (0, z) = 1+z
1-z-z 2 and we can recognize the generating function of the sequence {F i+2 } ∞ i=0 , the number of vertices of Γ n . Furthermore, ∂f (x,z)

∂x | x=1 = 2z (1-2z-2z 2 +z 3 ) 2 .
The generating function of the sequence {F i F i+1 } ∞ i=0 , the terms of which are known as golden rectangle numbers, is z/(1 -2z -2z 2 + z 3 ) [22, sequence A001654]. We have thus an alternative proof that the sequence

{W (Γ n )} ∞ n=0 is the convolution of {F i F i+1 } ∞ i=0 with {F i+1 F i+2 } ∞ i=0 .
A remarkable property of f (x, z) is that partial derivations are easy to compute which leads us to the following:

Corollary 4.2 Let k ≥ 1. The generating function of the sequence {f n,k } ∞ n=0 of number of pairs of vertices of Γ n at distance k is n≥0 f n,k z n = 2z(z + z 2 -z 3 ) k-1 (1 -z -z 2 ) k+1 .
Proof. By direct computation we have

∂f (x, z) ∂x = 2z (1 -z -z 2 + xz(-1 -z + z 2 )) 2 and ∂(1 -z -z 2 + xz(-1 -z + z 2 )) ∂x = z 3 -z 2 -z .
Therefore it is easy to prove by induction that for any k ≥ 1

∂ k f (x, z) ∂x k = 2(k!)z(z + z 2 -z 3 ) k-1 (1 -z -z 2 + xz(-1 -z + z 2 )) k+1 .
We have thus

∂ k f (x, z) ∂x k | x=0 = 2(k!)z(z + z 2 -z 3 ) k-1 (1 -z -z 2 ) k+1 .
On the other hand, from f (x, z) = n,k ≥0 f n,k x k z n we have for any fixed k,

∂ k f (x, z) ∂x k = n≥0   k!f n,k + k! k ≥k+1 f n,k x k -k   z n
The value for x = 0 implies the corollary.

We continue with Lucas cubes:

Theorem 4.3
The generating function of the sequence of ordered Hosoya polynomials for Λ n is

(x, z) = n≥0 H(Λ n , x)z n = 1 + 2x 2 z 3 -3x 2 z 4 + z 2 (1 + x) 2 1 -z -x 2 z 3 + x 2 z 4 -z 2 (1 + x) 2 .
To prove this result, let us first return to Fibonacci cubes. Let g n,k be the number of

(u, v) ∈ V (Γ n ) × V (Γ n ) such that d(u, v) = k where v begins with 0. For i, j ∈ {0, 1} let g ij n,k be the number of (u, v) ∈ V (Γ n ) × V (Γ n ) such that d(u, v) = k
where u ends with i, v begins with 0 and ends with j. Let us consider the generating polynomials g(x, z) = n,k≥0 g n,k x k z n and

g ij (x, z) = n,k≥0 g ij n,k x k z n .
Lemma 4.4 The generating functions of g n,k and g 01 n,k are

g(x, z) = 1 1 -z -z 2 + xz(-1 -z + z 2 ) -1 , g 01 (x, z) = xz 2 (1 -xz) (g(x, z) + 1) + x 2 z 2 1 -x 2 z 2 .
Proof. Assume throughout the proof that v begins with 0. In parallel to the general case we have g(x, z) = g 00 (x, z) + g 01 (x, z) + g 10 (x, z) + g 11 (x, z) .

(

) 6
Assume n ≥ 3. If u and v end with 1 then they are the concatenation of a string of F n-2

with 01, arbitrary for u, beginning with 0 for v. Thus g 11 n,k = g n-2,k . Using the initial values g 11 2,0 = 1, g 11 2,1 = g 11 2,2 = g 11 1,0 = g 11 1,1 = g 11 0,0 = g 0,0 = 0 we obtain

g 11 (x, z) = z 2 g(x, z) + z 2 . ( 7
)
Assume n ≥ 2. If u and v end with 0 then they are both the concatenation of Fibonacci strings of F n-1 with 0, arbitrary for u, beginning with 0 for v. Thus g 00 n,k = g n-1,k . Using the initial values g 00 1,0 = 1, g 00 1,1 = g 00 0,0 = g 0,0 = 0 we get

g 00 (x, z) = zg(x, z) + z . ( 8
)
Assume n ≥ 2. If u ends with 1 and v ends with 0, then u is the concatenation of a Fibonacci string of F 0 n-1 with 1, and v is the concatenation with 0 of a string of

F 0 n-1 ∪F 1 n-1 beginning with 0. Therefore g 10 n,k = g 00 n-1,k-1 + g 01 n-1,k-1 . By a symmetrical argument we have g 01 n,k = g 00 n-1,k-1 + g 10 n-1,k-1 . Therefore g 10 n,k + g 01 n,k = 2g 00 n-1,k-1 + g 10 n-1,k-1 + g 01 n-1,k-1 (n ≥ 2) (9)
and

g 10 n,k -g 01 n,k = -(g 10 n-1,k-1 -g 01 n-1,k-1 ) (n ≥ 2) . ( 10
)
The initial values being

g 10 1,1 = 1, g 01 0,1 = g 01 1,0 = g 01 0,0 = g 10 1,1 = g 10 0,0 = 0 . ( 11
)
Using these initial conditions and (9) we obtain by summation g 10 (x, z) + g 01 (x, z) -xz = 2xz 2 g(x, z) + 2xz 2 + xz(g 10 (x, z) + g 10 (x, z)) thus

g 10 (x, z) + g 01 (x, z) = 2xz 2 1 -xz (g(x, z) + 1) + xz 1 -xz . ( 12
)
By substitution in ( 6) of ( 7), ( 8) and ( 12) we obtain

g(x, z) = zg(x, z) + z 2 g(x, z) + 2xz 2 1 -xz (g(x, z) + 1) + xz 1 -xz + z + z 2
thus the expression of g(x, z).

¿From [START_REF] Dobrynin | Wiener index of hexagonal systems[END_REF] and the initial values [START_REF] Došlić | Counting conjugated circuits in benzenoid chains[END_REF] we obtain that g 01 n,k -g 10 n,k = 0 for k = n and g 01 n,n -g 10 n,n = (-1) n when n ≥ 1, therefore

g 01 (x, z) -g 10 (x, z) = -xz 1 + xz . ( 13
)
¿From ( 13), ( 12) and g 01 (x, z) = g 10 (x,z)+g 01 (x,z)+(g 01 (x,z)-g 10 (x,z))

2 we obtain the expression for

g 01 (x, z). Proof. (Theorem 4.3) Let n,k be the number of (u, v) ∈ V (Λ n ) × V (Λ n ) such that d(u, v) = k
and like previously consider ij n,k and the generating functions (x, z) and ij (x, z). We have again (x, z) = 1 + 00 (x, z) + 01 (x, z) + 10 (x, z) + 11 (x, z) .

(

) 14
If u and v end with 0 then u and v are arbitrary vertices of F 0 n and thus 00 (x, z) = f 00 (x, z) = zf (x, z) .

(

) 15
Assume n ≥ 2. If u and v end with 1 then they begin with 0 and thus u = 0u 01, v = 0v 01

where u and v are arbitrary strings of F 0 n-2 or the empty string if n = 2. Furthermore 11 n,k = 0 when n ≤ 1. We have thus 11 (x, z) = z 2 (f 00 (x, z) + 1) = z 3 f (x, z) + z 2 .

(

) 16
Assume n ≥ 2. If u ends with 0 and v ends with 1, then u is the concatenation with 0 of a string of F 0 n-1 ∪F 1 n-1 , and v is the concatenation with 1 of a string of F 0 n-1 beginning with 0. Therefore 01 n,k = g 00 n-1,k-1 + g 10 n-1,k-1 = g 01 n,k . Notice that, when n ≤ 1, 01 n,k = 0 = g 01 n,k , therefore 10 (x, z) = 01 (x, z) = g 01 (x, z) . [START_REF] Klavžar | Structure of Fibonacci cubes: a survey[END_REF] By substitution in ( 14) of ( 15), ( 16), [START_REF] Klavžar | Structure of Fibonacci cubes: a survey[END_REF] and using the values of f (x, z) (Theorem 4.1) and g 01 (x, z) (Lemma 4.4) we obtain the value of (x, z).

Notice that (0, z) = 1+z 2 1-z-z 2 and we can recognize the generating polynomial of V (Λ n ). Furthermore 1 2

∂ (x, z) ∂x | x=1 = z 2 4 -7z + 4z 2
(1 -2z -2z 2 + z 3 ) 2 is thus the generating function of W (Λ n ). The sequence {F i F i+2 } i=0 appears in [START_REF] Sloane | The On-Line Encyclopedia of Integer Sequences[END_REF] as sequence A059929 with generating function k(z) = i=0 F i F i+2 z i = (2z -z 2 )/((1+z)(1-3z +z 2 )). Then zk(z) = i=0 F i F i+2 z i+1 = (2z 2 -z 3 )/((1+z)(1-3z+z 2 )) whose derivate is i=1 iF i-1 F i+1 z i = z(4-7z+4z 2 ) (1-2z-2z 2 +z 3 ) 2 . We thus find an alternative proof that W (Λ n ) = nF n-1 F n+1 .

  0) | does not depend of i. So we may assume that i = 1. There are |F n-3 | Lucas strings whose first coordinate is 1, and there are |F n-1 | Lucas strings whose first coordinate is 0. Using Binet's formula and Theorem 3.4 we obtain a result for the average distance of Lucas cubes parallel to Fibonacci cubes:

Acknowledgements

We are grateful to Marko Petkovšek for directing us to the theory of Greene and Wilf from [12].

Work supported in part by the Research Grant P1-0297 of the Ministry of Higher Education, Science and Technology Slovenia and by the ANR Project Gratel, ANR-09-blan-0373-01.