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A015470 q-Fibonacci numbers for q=12, scaling a(n-2).

%I A015470 #14 Feb 03 2025 06:50:21
%S A015470 0,1,1,13,157,22621,3278173,5632106845,9794204234077,
%T A015470 201818365309759837,4211530365904119214429,
%U A015470 1041342647528423104910537053,260767900948768868884822059725149,773726564635922870118341112574642827613
%N A015470 q-Fibonacci numbers for q=12, scaling a(n-2).
%H A015470 Vincenzo Librandi, <a href="/A015470/b015470.txt">Table of n, a(n) for n = 0..60</a>
%F A015470 a(n) = a(n-1) + 12^(n-2)*a(n-2).
%p A015470 q:=12; 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 17 2019
%t A015470 RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]+a[n-2]*12^(n-2)},  a, {n, 60}] (* _Vincenzo Librandi_, Nov 09 2012 *)
%t A015470 F[n_, q_]:= Sum[QBinomial[n-j-1, j, q]*q^(j^2), {j, 0, Floor[(n-1)/2]}];
%t A015470 Table[F[n, 12], {n, 0, 20}] (* _G. C. Greubel_, Dec 17 2019 *)
%o A015470 (Magma) [0] cat[n le 2 select 1 else Self(n-1) + Self(n-2)*(12^(n-2)): n in [1..15]]; // _Vincenzo Librandi_, Nov 09 2012
%o A015470 (PARI) q=12; 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 17 2019
%o A015470 (Sage)
%o A015470 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 A015470 [F(n,12) for n in (0..20)] # _G. C. Greubel_, Dec 17 2019
%o A015470 (GAP) q:=12;; 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 17 2019
%Y A015470 q-Fibonacci numbers: A000045 (q=1), A015459 (q=2), A015460 (q=3), A015461 (q=4),
%Y A015470 A015462 (q=5), A015463 (q=6), A015464 (q=7), A015465 (q=8), A015467 (q=9), A015468 (q=10), A015469 (q=11), this sequence (q=12).
%K A015470 nonn,easy
%O A015470 0,4
%A A015470 _Olivier Gérard_