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%I #32 Sep 08 2022 08:44:40
%S 0,1,1,4,13,121,1174,30577,886423,67758322,5883579625,1339570631551,
%T 348759063908176,237649677731273173,185582515360156234789,
%U 379075929664916795231668,888014493839316022947740209
%N q-Fibonacci numbers for q=3.
%C Note that a(2,3,4,5) = {1,4,13,121} coincide with A003462(1,2,3,5). - _Zak Seidov_, Dec 19 2014
%H Vincenzo Librandi, <a href="/A015460/b015460.txt">Table of n, a(n) for n = 0..90</a>
%F a(n) = a(n-1) + 3^(n-2) * a(n-2).
%F Associated constant: C_3 = lim_{n->infinity} a(n)*a(n-2)/a(n-1)^2 = 1.147262608254535257774121586... . - _Benoit Cloitre_, Aug 30 2003
%F a(n)*a(n+3) - a(n)*a(n+2) - 3*a(n+1)*a(n+2) + 3*a(n+1)^2 = 0. - _Emanuele Munarini_, Dec 05 2017
%p q:=3; 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 RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]+a[n-2]*3^(n-2)}, a, {n, 30}] (* _Vincenzo Librandi_, Nov 08 2012 *)
%t F[n_, q_]:= Sum[QBinomial[n-j-1, j, q]*q^(j^2), {j, 0, Floor[(n-1)/2]}];
%t Table[F[n, 3], {n, 0, 20}] (* _G. C. Greubel_, Dec 16 2019 *)
%o (Magma) [0] cat[n le 2 select 1 else Self(n-1) + Self(n-2)*(3^(n-2)): n in [1..20]]; // _Vincenzo Librandi_, Nov 08 2012
%o (PARI) q=3; 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 (Sage)
%o 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 [F(n,3) for n in (0..20)] # _G. C. Greubel_, Dec 16 2019
%o (GAP) q:=3;; a:=[0,1];; for n in [3..30] do a[n]:=a[n-1]+q^(n-3)*a[n-2]; od; a; # _G. C. Greubel_, Dec 16 2019
%Y Cf. A003462.
%Y q-Fibonacci numbers: A000045 (q=1), A015459 (q=2), this sequence (q=3), A015461 (q=4), A015462 (q=5), A015463 (q=6), A015464 (q=7), A015465 (q=8), A015467 (q=9), A015468 (q=10), A015469 (q=11), A015470 (q=12).
%K nonn,easy
%O 0,4
%A _Olivier Gérard_
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