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A015474 q-Fibonacci numbers for q=3, scale a(n-1).

%I A015474 #16 Feb 03 2025 05:49:46
%S A015474 0,1,3,28,759,61507,14946960,10896395347,23830431570849,
%T A015474 156351472432735636,3077466055723967094237,
%U A015474 181721293280796005380336249,32191381943890636020834392595840,17107820211824904790829046440906141689
%N A015474 q-Fibonacci numbers for q=3, scale a(n-1).
%H A015474 Vincenzo Librandi, <a href="/A015474/b015474.txt">Table of n, a(n) for n = 0..60</a>
%F A015474 a(n) = 3^(n-1)*a(n-1) + a(n-2).
%p A015474 q:=3; 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 17 2019
%t A015474 RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]*3^(n-1) + a[n-2]},  a, {n, 20}] (* _Vincenzo Librandi_, Nov 09 2012 *)
%t A015474 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, 3], {n, 0, 20}] (* _G. C. Greubel_, Dec 17 2019 *)
%o A015474 (PARI) q=3; 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 17 2019
%o A015474 (Magma) q:=3; 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 17 2019
%o A015474 (Sage)
%o A015474 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 A015474 [F(n,3) for n in (0..20)] # _G. C. Greubel_, Dec 17 2019
%o A015474 (GAP) q:=3;; 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 17 2019
%Y A015474 q-Fibonacci numbers: A000045 (q=1), A015473 (q=2), this sequence (q=3), A015475 (q=4), A015476 (q=5), A015477 (q=6), A015479 (q=7), A015480 (q=8), A015481 (q=9), A015482 (q=10), A015484 (q=11), A015485 (q=12).
%Y A015474 Differs from A015460.
%K A015474 nonn,easy
%O A015474 0,3
%A A015474 _Olivier Gérard_