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A015463 q-Fibonacci numbers for q=6, scaling a(n-2).

%I A015463 #33 Feb 03 2025 06:46:30
%S A015463 0,1,1,7,43,1555,57283,12148963,2684744611,3403616850979,
%T A015463 4512743621400355,34305128668265064739,272902655183139496957219,
%U A015463 12446072589202949254455565603,594062125322746104949654522449187,162554939850629908283324416663519980835
%N A015463 q-Fibonacci numbers for q=6, scaling a(n-2).
%H A015463 Vincenzo Librandi, <a href="/A015463/b015463.txt">Table of n, a(n) for n = 0..70</a>
%F A015463 a(n) = a(n-1) + 6^(n-2)*a(n-2).
%F A015463 Associated constant: C_6 = lim_{n->infinity} a(n)*a(n-2)/a(n-1)^2 = 1.046607628427088904183396615... . - _Benoit Cloitre_, Aug 30 2003
%F A015463 a(n)*a(n+3) - a(n)*a(n+2) - 6*a(n+1)*a(n+2) + 6*a(n+1)^2 = 0. - _Emanuele Munarini_, Dec 05 2017
%p A015463 q:=6; 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 A015463 RecurrenceTable[{a[0]==0,a[1]==1,a[n]==a[n-1]+6^(n-2) a[n-2]},a,{n,20}] (* _Harvey P. Dale_, Nov 11 2011 *)
%t A015463 F[n_, q_]:= Sum[QBinomial[n-j-1, j, q]*q^(j^2), {j, 0, Floor[(n-1)/2]}];
%t A015463 Table[F[n, 6], {n, 0, 20}] (* _G. C. Greubel_, Dec 16 2019 *)
%o A015463 (Magma) [0] cat[n le 2 select 1 else Self(n-1) + Self(n-2)*(6^(n-2)): n in [1..20]]; // _Vincenzo Librandi_, Nov 09 2012
%o A015463 (PARI) q=6; 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 A015463 (Sage)
%o A015463 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 A015463 [F(n,6) for n in (0..20)] # _G. C. Greubel_, Dec 16 2019
%o A015463 (GAP) q:=6;; 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 A015463 q-Fibonacci numbers: A000045 (q=1), A015459 (q=2), A015460 (q=3), A015461 (q=4), A015462 (q=5), this sequence (q=6), A015464 (q=7), A015465 (q=8), A015467 (q=9), A015468 (q=10), A015469 (q=11), A015470 (q=12).
%K A015463 nonn,easy
%O A015463 0,4
%A A015463 _Olivier Gérard_