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A237420
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If n is odd, then a(n) = 0; otherwise, a(n) = n.
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9
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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
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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J. M. Arnaudiès, P. Delezoide et H. Fraysse, Exercices résolus d'Analyse du cours de mathématiques - 2, Dunod, Exercice 10, pp. 14-16.
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LINKS
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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).
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FORMULA
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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
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MAPLE
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seq(op([0, 2*i]), i=1..30); # Robert Israel, Aug 27 2015
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MATHEMATICA
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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 *)
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PROG
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(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
(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
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CROSSREFS
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Cf. A005843, A110660, A142150, A193356, A354280.
About the Cesàro mean theorem: A033999, A114112.
Sequence in context: A319813 A071648 A001613 * A130891 A091371 A144775
Adjacent sequences: A237417 A237418 A237419 * A237421 A237422 A237423
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KEYWORD
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nonn,easy
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AUTHOR
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Vincenzo Librandi, Feb 24 2014
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EXTENSIONS
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Edited by Bruno Berselli, Feb 27 2014
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STATUS
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approved
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