A237420 If n is odd, then a(n) = 0; otherwise, a(n) = n.

0, 0, 2, 0, 4, 0, 6, 0, 8, 0, 10, 0, 12, 0, 14, 0, 16, 0, 18, 0, 20, 0, 22, 0, 24, 0, 26, 0, 28, 0, 30, 0, 32, 0, 34, 0, 36, 0, 38, 0, 40, 0, 42, 0, 44, 0, 46, 0, 48, 0, 50, 0, 52, 0, 54, 0, 56, 0, 58, 0, 60, 0, 62, 0, 64, 0, 66, 0, 68, 0, 70, 0, 72, 0, 74

Offset: 0

Author

Vincenzo Librandi, Feb 24 2014

Keywords

• nonn
• easy

Comments

Normally the OEIS excludes sequences in which every other term is zero. But there are exceptions for especially important sequences like this one. - N. J. A. Sloane, Feb 27 2014

Essentially the factorial expansion of exp(-1): exp(-1) = Sum_{n>=1} a(n)/(n+1)!. - Joerg Arndt, Mar 13 2014

a(n) is the number of m < n for which a(m) has the same parity as n. For instance, a(4) = 4 because 4 has the same parity as a(0), a(1), a(2), and a(3). - Alec Jones, May 16 2016

This sequence is an example of a sequence that has no limit while the Cesàro means limit is infinite. See A354280 for further information. - Bernard Schott, May 22 2022

References

• J. M. Arnaudiès, P. Delezoide et H. Fraysse, Exercices résolus d'Analyse du cours de mathématiques - 2, Dunod, Exercice 10, pp. 14-16.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (corrected by Ray Chandler, Jan 19 2019).
• ProofWiki, Cesàro mean.
• Wikipedia, Ernesto Cesàro.
• Wikipédia, Lemme de Cesàro (in French).
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Crossrefs

Cf. A005843, A110660, A142150, A193356, A354280.

About the Cesàro mean theorem: A033999, A114112.

Programs

Magma

[IsOdd(n) select 0 else n: n in [1..80]];

Magma

[(1+(-1)^n)*n/2: n in [1..80]];

Magma

&cat [[n, 0]: n in [0..80 by 2]]; // Bruno Berselli, Nov 11 2016

Maple

seq(op([0,2*i]),i=1..30); # Robert Israel, Aug 27 2015

Mathematica

Table[If[OddQ[n], 0, n], {n, 80}]
CoefficientList[Series[2 x /(1 - x^2)^2, {x, 0, 80}], x]
LinearRecurrence[{0, 2, 0, -1}, {0, 0, 2, 0}, 75] (* Robert G. Wilson v, Nov 11 2016 *)
Riffle[Range[0,80,2],0] (* Harvey P. Dale, Mar 16 2021 *)

PARI

a(n)=if(n%2==0,n,0) \\ Anders Hellström, Aug 27 2015

Python

def a(n): return 0 if n%2 else n # Michael S. Branicky, Jun 05 2022

Formula

O.g.f.: 2*x^2/(1-x^2)^2.

E.g.f.: x*sinh(x). - Robert Israel, Aug 27 2015

a(n) = 2*a(n-2) - a(n-4) for n>4.

a(n) = 2*A142150(n) = (1+(-1)^n)*n/2 = n*((n-1) mod 2).

a(n) = floor(n^(-1)^n) for n>1. - Ilya Gutkovskiy, Aug 27 2015

Sum_{i=1..n} a(i) = A110660(n). - Bruno Berselli, Feb 27 2014

a(n) = -1 + ceiling((n + 1)^(sin(Pi*n/2) + cos(Pi*n))). - Lechoslaw Ratajczak, Nov 06 2016

Extensions

Edited by Bruno Berselli, Feb 27 2014