%I #15 Sep 08 2022 08:44:40

%S 0,1,8,513,262664,1075872257,35254182380040,9241672386909078017,

%T 19381191729586400963887624,325162439984693881306137776652801,

%U 43642563925681986905603214423711358943752

%N q-Fibonacci numbers for q=8.

%H Vincenzo Librandi, <a href="/A015480/b015480.txt">Table of n, a(n) for n = 0..40</a>

%F a(n) = 8^(n-1)*a(n-1) + a(n-2).

%p q:=8; seq(add((product((1-q^(2*(n-j-1-k)))/(1-q^(2*k+2)), k=0..j-1))* q^binomial(n-2*j,2), j = 0..floor((n-1)/2)), n = 0..20); # _G. C. Greubel_, Dec 18 2019

%t RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]*8^(n-1)+a[n-2]}, a, {n, 20}] (* _Vincenzo Librandi_, Nov 10 2012 *)

%t F[n_, q_]:= Sum[QBinomial[n-j-1, j, q^2]*q^Binomial[n-2*j,2], {j, 0, Floor[(n-1)/2]}]; Table[F[n, 8], {n, 0, 20}] (* _G. C. Greubel_, Dec 18 2019 *)

%o (PARI) q=8; m=20; v=concat([0,1], vector(m-2)); for(n=3, m, v[n]=q^(n-2)*v[n-1]+v[n-2]); v \\ _G. C. Greubel_, Dec 18 2019

%o (Magma) q:=8; I:=[0,1]; [n le 2 select I[n] else q^(n-2)*Self(n-1) + Self(n-2): n in [1..20]]; // _G. C. Greubel_, Dec 18 2019

%o (Sage)

%o def F(n,q): return sum( q_binomial(n-j-1, j, q^2)*q^binomial(n-2*j,2) for j in (0..floor((n-1)/2)))

%o [F(n,8) for n in (0..20)] # _G. C. Greubel_, Dec 18 2019

%o (GAP) q:=8;; a:=[0,1];; for n in [3..20] do a[n]:=q^(n-2)*a[n-1]+a[n-2]; od; a; # _G. C. Greubel_, Dec 18 2019

%Y q-Fibonacci numbers: A000045 (q=1), A015473 (q=2), A015474 (q=3), A015475 (q=4), A015476 (q=5), A015477 (q=6), A015479 (q=7), this sequence (q=8), A015481 (q=9), A015482 (q=10), A015484 (q=11), A015485 (q=12).

%Y Differs from A015465.

%K nonn,easy

%O 0,3

%A _Olivier Gérard_