A181560 a(n+1) = a(n-1) + 2 a(n-2) - a(n-4) ; a(0)=1, a(n)=0 for 0 < n < 5;

1, 0, 0, 0, 0, -1, 0, -1, -2, -1, -3, -5, -4, -9, -13, -14, -26, -36, -45, -75, -103, -139, -217, -300, -420, -631, -881, -1254, -1843, -2596, -3720, -5401, -7658, -10998, -15864, -22594, -32459, -46664, -66649, -95718, -137383, -196557, -282155

Offset

0,9

Comments

a(n) is the constant term of the canonical representative (polynomial of degree < 5) of x^n (mod x^5-x^3-2*x^2+1), see example.

Links

Table of n, a(n) for n=0..42.

Index entries for linear recurrences with constant coefficients, signature (0,1,2,0,-1).

Formula

G.f.: sum( a(k) x^k, k=0...oo ) = (1 - x^2 - 2*x^3)/(1 - x^2 - 2*x^3 + x^5)

Example

x^6 = x^4 + 2*x^3 - x (mod x^5 - x^3 - 2*x^2 + 1), and the l.h.s. has no constant term, so a(6) = 0.
x^14 = 14*x^4 + 26*x^3 + 22*x^2 - 9*x - 13 (mod x^5 - x^3 - 2*x^2 + 1), and the constant term on the r.h.s. is a(14) = -13.

Mathematica

LinearRecurrence[{0, 1, 2, 0, -1}, {1, 0, 0, 0, 0}, 50] (* Harvey P. Dale, Sep 21 2024 *)

Prog

(PARI) a(n) = polcoeff( lift( Mod( x, x^5-x^3-2*x^2+1)^n), 0)

Crossrefs

Sequence in context: A113790 A181094 A282666 * A250103 A063705 A184250

Adjacent sequences: A181557 A181558 A181559 * A181561 A181562 A181563

Keyword

easy,sign,changed

Author

M. F. Hasler, Nov 04 2010

Status

approved