A239229 Euler characteristic of n-holed torus: 2 - 2*n.

2, 0, -2, -4, -6, -8, -10, -12, -14, -16, -18, -20, -22, -24, -26, -28, -30, -32, -34, -36, -38, -40, -42, -44, -46, -48, -50, -52, -54, -56, -58, -60, -62, -64, -66, -68, -70, -72, -74, -76, -78, -80, -82, -84, -86, -88, -90, -92, -94, -96, -98, -100, -102

Offset

0,1

References

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Plane Geometry), p. 450.

James Munkres, Topology, 2nd ed., Pearson, 2000.

David S. Richeson, Euler's Gem: The Polyhedron Formula and the Birth of Topology, Princeton University Press, 2008. See pp. 177-183.

Links

Table of n, a(n) for n=0..52.

Richard Courant and Herbert Robbins, What Is Mathematics?, Oxford, 1941, pp. 258-259.

Eric Weisstein's World of Mathematics, Euler Characteristic.

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

Formula

a(n) = 2 - 2*n.

G.f.: 2*(1-2*x)/(1-x)^2. - Vincenzo Librandi, Feb 01 2015

a(n) = 2*A022958(n+1). - R. J. Mathar, Oct 05 2017

E.g.f.: 2*exp(x)*(1 - x). - Stefano Spezia, Sep 10 2022

Maple

A239229:=n->2 - 2*n; seq(A239229(n), n=0..100); # Wesley Ivan Hurt, Mar 15 2014

Mathematica

Table[2 - 2 n, {n, 0, 100}] (* Wesley Ivan Hurt, Mar 15 2014 *)
(* Alternative: *)
LinearRecurrence[{2, -1}, {2, 0}, 80] (* Harvey P. Dale, Jan 31 2015 *)
(* Alternative: *)
CoefficientList[Series[2 (1-2 x) / (1 - x)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Feb 01 2015 *)

Prog

(Magma) [2-2*n: n in [0..60]]; // Vincenzo Librandi, Feb 01 2015

Crossrefs

Cf. A010673, A022958.

Sequence in context: A194751 A194739 A194765 * A058210 A382867 A360860

Adjacent sequences: A239226 A239227 A239228 * A239230 A239231 A239232

Keyword

sign,easy,changed

Author

Eric M. Schmidt, Mar 12 2014

Status

approved