A015480 q-Fibonacci numbers for q=8, scaling a(n-1).

0, 1, 8, 513, 262664, 1075872257, 35254182380040, 9241672386909078017, 19381191729586400963887624, 325162439984693881306137776652801, 43642563925681986905603214423711358943752

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..40

Crossrefs

q-Fibonacci numbers: A000045 (q=1), A015473 (q=2), A015474 (q=3), A015475 (q=4), A015476 (q=5), A015477 (q=6), A015479 (q=7), this sequence (q=8), A015481 (q=9), A015482 (q=10), A015484 (q=11), A015485 (q=12).

Differs from A015465.

Programs

GAP

q:=8;; 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 18 2019

Magma

q:=8; 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 18 2019

Maple

q:=8; 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 18 2019

Mathematica

RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]*8^(n-1)+a[n-2]}, a, {n, 20}] (* Vincenzo Librandi, Nov 10 2012 *)
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, 8], {n, 0, 20}] (* G. C. Greubel, Dec 18 2019 *)

PARI

q=8; 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 18 2019

Sage

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)))
[F(n,8) for n in (0..20)] # G. C. Greubel, Dec 18 2019

Formula

a(n) = 8^(n-1)*a(n-1) + a(n-2).