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

0, 1, 1, 7, 43, 1555, 57283, 12148963, 2684744611, 3403616850979, 4512743621400355, 34305128668265064739, 272902655183139496957219, 12446072589202949254455565603, 594062125322746104949654522449187, 162554939850629908283324416663519980835

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..70

Crossrefs

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).

Programs

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

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

Maple

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

Mathematica

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 *)
F[n_, q_]:= Sum[QBinomial[n-j-1, j, q]*q^(j^2), {j, 0, Floor[(n-1)/2]}];
Table[F[n, 6], {n, 0, 20}] (* G. C. Greubel, Dec 16 2019 *)

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

Sage

def F(n,q): return sum( q_binomial(n-j-1, j, q)*q^(j^2) for j in (0..floor((n-1)/2)))
[F(n,6) for n in (0..20)] # G. C. Greubel, Dec 16 2019

Formula

a(n) = a(n-1) + 6^(n-2)*a(n-2).

Associated constant: C_6 = lim_{n->infinity} a(n)*a(n-2)/a(n-1)^2 = 1.046607628427088904183396615... . - Benoit Cloitre, Aug 30 2003

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