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A112576 A Chebyshev-related transform of the Fibonacci numbers. 4
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 (list; graph; refs; listen; history; text; internal format)
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)).

REFERENCES

D. Huylebrouck, The Meta-Golden Ratio Chi, Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture, http://archive.bridgesmathart.org/2014/bridges2014-151.pdf

LINKS

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

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

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Last modified February 27 13:18 EST 2020. Contains 332306 sequences. (Running on oeis4.)