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

Author

Roger L. Bagula, May 30 2005

Keywords

• sign
• easy
• less

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).

Crossrefs

Cf. A116415.

Programs

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

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 *)

Sage

[-3^((n+1)/2)*chebyshev_U(n-1, 5/(2*sqrt(3))) for n in (0..30)] # G. C. Greubel, Sep 10 2021

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

Extensions

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

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