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A010673 Period 2: repeat [0, 2]. 20
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 (list; graph; refs; listen; history; text; internal format)
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
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
PadRight[{}, 120, {0, 2}] (* or *) LinearRecurrence[{0, 1}, {0, 2}, 120] (* Harvey P. Dale, May 29 2016 *)
PROG
(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
(GAP) Flat(List([0..80], n->[0, 2])); # Muniru A Asiru, Oct 26 2018
CROSSREFS
Cf. A109613.
Sequence in context: A267602 A021499 A176742 * A084099 A036665 A053472
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified May 29 00:29 EDT 2024. Contains 372921 sequences. (Running on oeis4.)