A010673 - OEIS

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

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OFFSET

0,2

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

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).

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

MAPLE

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

MATHEMATICA