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A015467 q-Fibonacci numbers for q=9, scaling a(n-2).

%I A015467 #16 Feb 03 2025 06:47:28
%S A015467 0,1,1,10,91,7381,604432,436445101,321656391613,2087825044676482,
%T A015467 13848340772676227455,808880048095782179467153,
%U A015467 48286987465947852695801396608,25383561292811993463191359951919785,13637696871632801620185917930189837576233
%N A015467 q-Fibonacci numbers for q=9, scaling a(n-2).
%H A015467 Vincenzo Librandi, <a href="/A015467/b015467.txt">Table of n, a(n) for n = 0..60</a>
%F A015467 a(n) = a(n-1) + 9^(n-2) * a(n-2).
%p A015467 q:=9; 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 A015467 RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1] + a[n-2]*9^(n-2)},  a, {n, 20}] (* _Vincenzo Librandi_, Nov 09 2012 *)
%t A015467 F[n_, q_]:= Sum[QBinomial[n-j-1, j, q]*q^(j^2), {j, 0, Floor[(n-1)/2]}];
%t A015467 Table[F[n, 9], {n, 0, 20}] (* _G. C. Greubel_, Dec 16 2019 *)
%o A015467 (Magma) [0] cat[n le 2 select 1 else Self(n-1) + Self(n-2)*(9^(n-2)): n in [1..15]]; // _Vincenzo Librandi_, Nov 09 2012
%o A015467 (PARI) q=9; 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 A015467 (Sage)
%o A015467 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 A015467 [F(n,9) for n in (0..20)] # _G. C. Greubel_, Dec 16 2019
%o A015467 (GAP) q:=9;; 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 A015467 q-Fibonacci numbers: A000045 (q=1), A015459 (q=2), A015460 (q=3), A015461 (q=4), A015462 (q=5), A015463 (q=6), A015464 (q=7), A015465 (q=8), this sequence (q=9), A015468 (q=10), A015469 (q=11), A015470 (q=12).
%K A015467 nonn,easy
%O A015467 0,4
%A A015467 _Olivier Gérard_