A078046 Expansion of (1-x)/(1 + x + x^2 - x^3).

1, -2, 1, 2, -5, 4, 3, -12, 13, 2, -27, 38, -9, -56, 103, -56, -103, 262, -215, -150, 627, -692, -85, 1404, -2011, 522, 2893, -5426, 3055, 5264, -13745, 11536, 7473, -32754, 36817, 3410, -72981, 106388, -29997, -149372, 285757, -166382, -268747, 720886, -618521, -371112, 1710519, -1957928

Offset: 0

Author

N. J. A. Sloane, Nov 17 2002

Keywords

• sign
• easy

Comments

The root of the denominator [1 + x + x^2 - x^3] is the tribonacci constant.

This is the negative of the tribonacci numbers, signature (0, 1, 0), in reverse order, starting from A001590(-1), going backwards A001590(-2), A001590(-3), ... - Peter M. Chema, Dec 31 2016

Examples

			G.f. = 1 - 2*x + x^2 + 2*x^3 - 5*x^4 + 4*x^5 + 3*x^6 - 12*x^7 + 13*x^8 + ...
		

Links

• Benjamin Braun, W. K. Hough, Matching and Independence Complexes Related to Small Grids, arXiv preprint arXiv:1606.01204 [math.CO], 2016.
• Wesley K. Hough, On Independence, Matching, and Homomorphism Complexes, (2017), Theses and Dissertations--Mathematics, 42.
• Index entries for linear recurrences with constant coefficients, signature (-1,-1,1).

Crossrefs

First differences of A057597.

Cf. A001590.

Programs

Mathematica

a[ n_] := If[ n >= 0, SeriesCoefficient[ (1 - x) / (1 + x + x^2 - x^3), {x, 0, n}], SeriesCoefficient [ -x^2 (1 - x) / (1 - x - x^2 - x^3),{x, 0, -n}]]; (* Michael Somos, Jun 01 2014 *)

PARI

Vec((1-x)/(1+x+x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

PARI

{a(n) = if( n>=0, polcoeff( (1 - x) / (1 + x + x^2 - x^3) + x * O(x^n), n), polcoeff( -x^2 * (1 - x) / (1 - x - x^2 - x^3) + x * O(x^-n), -n))}; /* Michael Somos, Jun 01 2014 */

Formula

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

Recurrence: a(n) = a(n-3) - a(n-2) - a(n-1) for n > 2.

a(-1 - n) = - A001590(n). - Michael Somos, Jun 01 2014