A010673 Period 2: repeat [0, 2].

0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Euler number (or Euler characteristic) of (n+1)-sphere. - Franz Vrabec, Sep 07 2007

First differences of A109613. - Reinhard Zumkeller, Dec 05 2009

a(n) = Sum_{k=0..n-1} (-1)^k*N_k, for n >= 1, is Schläfli's generalization of Euler's formula for simply-connected n-dimensional polytopes. N_0 is the number of vertices, ..., N_{d-1} is the number of (d-1)-dimensional faces. See Coxeter's book for references, also for Poincaré's proof. - Wolfdieter Lang, Feb 09 2018

Decimal expansion of 2/99. - R. J. Mathar, May 15 2025

References

• R. Carter, G. Segal, I. Macdonald, Lectures on Lie Groups and Lie Algebras, London Mathematical Society Student Texts 32, Cambridge University Press, 1995; see p. 68.
• H. S. M. Coxeter, Regular Polytopes, third ed., Dover publications, New York, 1973, p. 165.

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,1).

Crossrefs

Cf. A109613.

Programs

GAP

Flat(List([0..80],n->[0,2])); # Muniru A Asiru, Oct 26 2018

Maple

seq(op([0,2]),n=0..80); # Muniru A Asiru, Oct 26 2018

Mathematica

PadRight[{},120,{0,2}] (* or *) LinearRecurrence[{0,1},{0,2},120] (* Harvey P. Dale, May 29 2016 *)

Maxima

makelist(if evenp(n) then 0 else 2, n, 0, 30); /* Martin Ettl, Nov 11 2012 */

Maxima

makelist(concat(0,", ",2), n, 0, 40); /* Bruno Berselli, Nov 13 2012 */

PARI

a(n)=1-(-1)^n \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = 1 - (-1)^n.

a(n) = 2*(n mod 2). - Paolo P. Lava, Oct 20 2006

G.f.: -2*x / ((x-1)*(1+x)). - R. J. Mathar, Apr 06 2011

E.g.f.: (exp(2*x) - 1)/exp(x). - Elmo R. Oliveira, Dec 19 2023