A078046 - OEIS

%I #32 Oct 21 2017 21:00:05

%S 1,-2,1,2,-5,4,3,-12,13,2,-27,38,-9,-56,103,-56,-103,262,-215,-150,

%T 627,-692,-85,1404,-2011,522,2893,-5426,3055,5264,-13745,11536,7473,

%U -32754,36817,3410,-72981,106388,-29997,-149372,285757,-166382,-268747,720886,-618521,-371112,1710519,-1957928

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

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

%C 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

%H Benjamin Braun, W. K. Hough, <a href="https://arxiv.org/abs/1606.01204">Matching and Independence Complexes Related to Small Grids</a>, arXiv preprint arXiv:1606.01204 [math.CO], 2016.

%H Wesley K. Hough, <a href="https://dx.doi.org/10.13023/ETD.2017.119">On Independence, Matching, and Homomorphism Complexes</a>, (2017), Theses and Dissertations--Mathematics, 42.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-1,-1,1).

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

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

%F a(-1 - n) = - A001590(n). - _Michael Somos_, Jun 01 2014

%e 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 + ...

%t 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 *)

%o (PARI) Vec((1-x)/(1+x+x^2-x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012

%o (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 */

%Y First differences of A057597.

%Y Cf. A001590.

%K sign,easy

%O 0,2

%A _N. J. A. Sloane_, Nov 17 2002