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A015465 q-Fibonacci numbers for q=8, scaling a(n-2).

%I A015465 #16 Feb 03 2025 06:47:11
%S A015465 0,1,1,9,73,4681,303689,153690697,79763939913,322392516534857,
%T A015465 1338539241447957065,43272129632752387301961,
%U A015465 1437288838737538572434088521,371706200490726725394268777423433,98770108622737228265012391281001570889
%N A015465 q-Fibonacci numbers for q=8, scaling a(n-2).
%H A015465 Vincenzo Librandi, <a href="/A015465/b015465.txt">Table of n, a(n) for n = 0..60</a>
%F A015465 a(n) = a(n-1) + 8^(n-2) * a(n-2).
%p A015465 q:=8; seq(add((product((1-q^(n-j-1-k))/(1-q^(k+1)), k=0..j-1))*q^(j^2), j = 0..floor((n-1)/2)), n = 0..20); # _G. C. Greubel_, Dec 16 2019
%t A015465 RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]+a[n-2]*8^(n-2)},  a, {n, 20}] (* _Vincenzo Librandi_, Nov 09 2012 *)
%t A015465 F[n_, q_]:= Sum[QBinomial[n-j-1, j, q]*q^(j^2), {j, 0, Floor[(n-1)/2]}];
%t A015465 Table[F[n, 8], {n, 0, 20}] (* _G. C. Greubel_, Dec 16 2019 *)
%o A015465 (Magma) [0] cat[n le 2 select 1 else Self(n-1) + Self(n-2)*(8^(n-2)): n in [1..15]]; // _Vincenzo Librandi_, Nov 09 2012
%o A015465 (PARI) q=8; m=20; v=concat([0,1], vector(m-2)); for(n=3, m, v[n]=v[n-1]+q^(n-3)*v[n-2]); v \\ _G. C. Greubel_, Dec 16 2019
%o A015465 (Sage)
%o A015465 def F(n,q): return sum( q_binomial(n-j-1, j, q)*q^(j^2) for j in (0..floor((n-1)/2)))
%o A015465 [F(n,8) for n in (0..20)] # _G. C. Greubel_, Dec 16 2019
%o A015465 (GAP) q:=8;; a:=[0,1];; for n in [3..20] do a[n]:=a[n-1]+q^(n-3)*a[n-2]; od; a; # _G. C. Greubel_, Dec 16 2019
%Y A015465 q-Fibonacci numbers: A000045 (q=1), A015459 (q=2), A015460 (q=3), A015461 (q=4), A015462 (q=5), A015463 (q=6), A015464 (q=7), this sequence (q=8), A015467 (q=9), A015468 (q=10), A015469 (q=11), A015470 (q=12).
%K A015465 nonn,easy
%O A015465 0,4
%A A015465 _Olivier Gérard_