A078042 Expansion of (1-x)/(1+x-x^2+x^3).

1, -2, 3, -6, 11, -20, 37, -68, 125, -230, 423, -778, 1431, -2632, 4841, -8904, 16377, -30122, 55403, -101902, 187427, -344732, 634061, -1166220, 2145013, -3945294, 7256527, -13346834, 24548655, -45152016, 83047505, -152748176, 280947697, -516743378, 950439251, -1748130326, 3215312955

Offset

0,2

Comments

Absolute values give coordination sequence for (3,infinity,infinity) tiling of hyperbolic plane. - N. J. A. Sloane, Dec 29 2015

a(n) is the upper left entry of the n-th power of the 3 X 3 matrix M = [-2, -2, 1; 1, 1, 0; 1, 0, 0]; a(n) = M^n [1, 1]. - Philippe Deléham, Apr 19 2023

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

J. W. Cannon and P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.

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

Formula

a(n) = -a(n-1) + a(n-2) - a(n-3) for n > 2; a(0)=1, a(1)=-2, a(2)=3. - Harvey P. Dale, Jun 01 2012

a(n) = (-1)^n * A001590(n+2).

a(n) = Sum_{k=0..n} A188316(n,k)*(-2)^k. - Philippe Deléham, Apr 19 2023

Mathematica

CoefficientList[Series[(1-x)/(1+x-x^2+x^3), {x, 0, 40}], x] (* or *) LinearRecurrence[{-1, 1, -1}, {1, -2, 3}, 40] (* Harvey P. Dale, Jun 01 2012 *)

Prog

(PARI) Vec((1-x)/(1+x-x^2+x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012
(Magma) [n le 3 select -n*(-1)^n else -Self(n-1)+Self(n-2)-Self(n-3): n in [1..50]]; // Vincenzo Librandi, Dec 30 2015

Crossrefs

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.

Sequence in context: A335628 A355560 A001590 * A047081 A115792 A054177

Adjacent sequences: A078039 A078040 A078041 * A078043 A078044 A078045

Keyword

sign,easy

Author

N. J. A. Sloane, Nov 17 2002

Status

approved