A015473 q-Fibonacci numbers for q=2, scale a(n-1).

0, 1, 2, 9, 74, 1193, 38250, 2449193, 313534954, 80267397417, 41097221012458, 42083634584154409, 86187324725569242090, 353023324159566199755049, 2891967157702491033962603498, 47381990264820937260009495466281

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

Comments

a(1) = 1, a(n+1) = denominator of continued fraction [1;2,4,8,...,2^n]. - Amarnath Murthy, May 02 2001

The difference equation y(n, x, s) = q^(n-1)*x*y(n-1, x, s) + s*y(n-2, x, s) yields a type of two variable q-Fibonacci polynomials in the form F(n, x, s, q) = Sum_{j=0..floor((n-1)/2)} q-binomial(n-j-1,j, q^2)*q^binomial(n-2*j,2)* x^(n-2*j)*s^j. When x=s=1 these polynomials reduce to q-Fibonacci numbers. This family of q-Fibonacci numbers is different from that of the q-Fibonacci numbers defined in A015459. - G. C. Greubel, Dec 17 2019

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..80

Crossrefs

Cf. A061377.

q-Fibonacci numbers: A000045 (q=1), this sequence (q=2), A015474 (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).

Programs

GAP

q:=2;; 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

Magma

[0] cat [n le 2 select n else 2^(n-1)*Self(n-1) + Self(n-2): n in [1..16]]; // Vincenzo Librandi, Nov 09 2012

Maple

q:=2; 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

Mathematica

RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]*2^(n-1)+a[n-2]},  a, {n, 30}] (* Vincenzo Librandi, Nov 09 2012 *)
Join[{0},Denominator[Table[FromContinuedFraction[2^Range[0,n]],{n,0,20}]]] (* Harvey P. Dale, Feb 09 2013 *)
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, 2], {n, 0, 20}] (* G. C. Greubel, Dec 17 2019 *)

PARI

q=2; 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

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,2) for n in (0..20)] # G. C. Greubel, Dec 17 2019

Formula

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