A112576 A Chebyshev-related transform of the Fibonacci numbers.

0, 1, 1, 4, 6, 16, 29, 67, 132, 288, 588, 1253, 2597, 5480, 11430, 24020, 50233, 105383, 220632, 462528, 968808, 2030377, 4253641, 8913436, 18675174, 39131464, 81989909, 171795691, 359958780, 754224480, 1580315220, 3311234189

Offset

0,4

Comments

Transform of the Fibonacci numbers by the Chebyshev related transform which maps g(x) -> (1/(1-x^2))g(x/(1-x^2)).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

D. Huylebrouck, The Meta-Golden Ratio Chi, Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture.

Index entries for linear recurrences with constant coefficients, signature (1,3,-1,-1).

Formula

G.f.: x/(1-x-3*x^2+x^3+x^4).

a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*F(n-2*k).

a(n) = Sum_{k=0..n} C((n+k)/2, k)*(1+(-1)^(n-k))*F(k)/2.

a(n) = (Fibonacci(n+1, (1+sqrt(5))/2) - Fibonacci(n+1, (1-sqrt(5))/2) )/sqrt(5), where Fibonacci(n,x) is the Fibonacci polynomial (see A011973). - G. C. Greubel, Jul 29 2019

Mathematica

(* see A192232 for Mmca code. - M. F. Hasler, Apr 05 2016 *)

Prog

(PARI) Vec(x/(1-x-3*x^2+x^3+x^4)+O(x^40)) \\ M. F. Hasler, Apr 05 2016
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x/(1-x-3*x^2+x^3+x^4) )); // G. C. Greubel, Jul 29 2019
(Sage) (x/(1-x-3*x^2+x^3+x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jul 29 2019
(GAP) a:=[0, 1, 1, 4];; for n in [5..40] do a[n]:=a[n-1]+3*a[n-2]-a[n-3] -a[n-4]; od; a; # G. C. Greubel, Jul 29 2019

Crossrefs

Cf. A011973, A192232.

Sequence in context: A261682 A102731 A007179 * A174804 A081487 A099430

Adjacent sequences: A112573 A112574 A112575 * A112577 A112578 A112579

Keyword

easy,nonn

Author

Paul Barry, Sep 14 2005

Status

approved