A078058 Expansion of (1-x)/(1+2*x-x^2+x^3).

1, -3, 7, -18, 46, -117, 298, -759, 1933, -4923, 12538, -31932, 81325, -207120, 527497, -1343439, 3421495, -8713926, 22192786, -56520993, 143948698, -366611175, 933692041, -2377943955, 6056191126, -15424018248, 39282171577, -100044552528, 254795294881, -648917313867

Offset: 0

Author

N. J. A. Sloane, Nov 17 2002

Keywords

• sign
• easy

Comments

a(n) is the upper left entry of the n-th power of the 3 X 3 matrix M = [-3, -3, 1; 1, 1, 0; 1, 0, 0]; a(n) = M^n [1, 1]. - Philippe Deléham, Apr 19 2023

Links

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

Programs

Mathematica

CoefficientList[Series[(1-x)/(1+2x-x^2+x^3),{x,0,30}],x] (* or *) LinearRecurrence[{-2,1,-1},{1,-3,7},31] (* Harvey P. Dale, Oct 22 2011 *)

PARI

Vec((1-x)/(1+2*x-x^2+x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

Formula

a(n) = -2*a(n-1) + a(n-2) - a(n-3) for n > 2; a(0) = 1, a(1) = -3, a(2) = 7. - Harvey P. Dale, Oct 22 2011

a(n) = Sum_{k = 0..n} A188316(n, k)*(-3)^k. - Philippe Deléham, Apr 19 2023

a(n) = A077986(n)-A077986(n-1) . - R. J. Mathar, Mar 19 2025