A015462 q-Fibonacci numbers for q=5, scaling a(n-2).

0, 1, 1, 6, 31, 781, 20156, 2460781, 317398281, 192565913906, 124176269429531, 376229476867085781, 1213035110624630757656, 18371792960261297531148281, 296169521847801754865890523281, 22426801247965814514582357345601406

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), this sequence (q=5), A015463 (q=6), A015464 (q=7), A015465 (q=8), A015467 (q=9), A015468 (q=10), A015469 (q=11), A015470 (q=12).

Programs

GAP

q:=5;; 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)*(5^(n-2)): n in [1..20]]; // Vincenzo Librandi, Nov 09 2012

Maple

q:=5; 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]+a[n-2]*5^(n-2)},  a, {n, 20}] (* Vincenzo Librandi, Nov 09 2012 *)
nxt[{n_,a_,b_}]:={n+1,b,b+a*5^(n-1)}; NestList[nxt,{1,0,1},20][[All,2]] (* Harvey P. Dale, Aug 19 2019 *)
F[n_, q_]:= Sum[QBinomial[n-j-1, j, q]*q^(j^2), {j, 0, Floor[(n-1)/2]}];
Table[F[n, 5], {n, 0, 20}] (* G. C. Greubel, Dec 16 2019 *)

PARI

q=5; 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,5) for n in (0..20)] # G. C. Greubel, Dec 16 2019

Formula

a(n) = a(n-1) + 5^(n-2)*a(n-2).

Associated constant: C_5 = lim_{n->oo} a(n)*a(n-2)/a(n-1)^2 = 1.064478080430862119874641125... . - Benoit Cloitre, Aug 30 2003

a(n)*a(n+3) - a(n)*a(n+2) - 5*a(n+1)*a(n+2) + 5*a(n+1)^2 = 0. - Emanuele Munarini, Dec 05 2017