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

Author

Paul Barry, Sep 14 2005

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A011973, A192232.

Programs

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

Magma

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x/(1-x-3*x^2+x^3+x^4) )); // G. C. Greubel, Jul 29 2019

Mathematica

(* see A192232 for Mmca code. - M. F. Hasler, Apr 05 2016 *)
LinearRecurrence[{1,3,-1,-1},{0,1,1,4},40] (* Harvey P. Dale, May 09 2025 *)

PARI

Vec(x/(1-x-3*x^2+x^3+x^4)+O(x^40)) \\ M. F. Hasler, Apr 05 2016

Sage

(x/(1-x-3*x^2+x^3+x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jul 29 2019

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