A106732 Expansion of -3*x/(1 - 5*x + 3*x^2).

0, -3, -15, -66, -285, -1227, -5280, -22719, -97755, -420618, -1809825, -7787271, -33506880, -144172587, -620342295, -2669193714, -11484941685, -49417127283, -212630811360, -914902674951, -3936620940675, -16938396678522, -72882120570585

Offset

0,2

Links

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

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (5,-3).

Formula

G.f.: -3*x/(1 - 5*x + 3*x^2).

a(n) = 5*a(n-1) - 3*a(n-2), a(0) = 0, a(1) = -3.

a(n) = -3*A116415(n-1), n>0.

a(n) = -3^((n+1)/2)*ChebyshevU(n-1, 5/(2*sqrt(3))). - G. C. Greubel, Sep 10 2021

Maple

a[0]:=0: a[1]:=-3: for n from 2 to 22 do a[n]:=5*a[n-1]-3*a[n-2] od: seq(a[n], n=0..30);

Mathematica

LinearRecurrence[{5, -3}, {0, -3}, 31] (* G. C. Greubel, Sep 10 2021 *)
CoefficientList[Series[-3x/(1-5x+3x^2), {x, 0, 30}], x] (* Harvey P. Dale, Jan 29 2025 *)

Prog

(Magma) [n le 2 select -3*(1+(-1)^n)/2 else 5*Self(n-1) - 3*Self(n-2): n in [1..31]]; // G. C. Greubel, Sep 10 2021
(Sage) [-3^((n+1)/2)*chebyshev_U(n-1, 5/(2*sqrt(3))) for n in (0..30)] # G. C. Greubel, Sep 10 2021

Crossrefs

Cf. A116415.

Sequence in context: A098102 A144067 A001447 * A052981 A086200 A325586

Adjacent sequences: A106729 A106730 A106731 * A106733 A106734 A106735

Keyword

sign,easy,less,changed

Author

Roger L. Bagula, May 30 2005

Extensions

Edited by N. J. A. Sloane, Apr 30 2006

New name by G. C. Greubel, Sep 10 2021

Status

approved