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A354280
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a(n) is the numerator of Cesàro means sequence c(n) of A237420 when the denominator is A141310(n).
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2
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0, 0, 2, 1, 6, 2, 12, 3, 20, 4, 30, 5, 42, 6, 56, 7, 72, 8, 90, 9, 110, 10, 132, 11, 156, 12, 182, 13, 210, 14, 240, 15, 272, 16, 306, 17, 342, 18, 380, 19, 420, 20, 462, 21, 506, 22, 552, 23, 600, 24, 650, 25, 702, 26, 756, 27, 812, 28, 870, 29, 930, 30, 992, 31, 1056, 32, 1122, 33, 1190
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OFFSET
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0,3
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COMMENTS
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So, we get c(n) = a(n) / A141310(n) for n >= 0 (see Formula and Example section).
Cesàro mean theorem: when the series u(n) has a limit (finite or infinite) in the usual sense, then c(n) = (u(0)+...+u(n))/(n+1) has the same Cesàro limit, but the converse is false.
A237420 is such a counterexample in the case of an infinite limit.
Proof: A237420 is not convergent in the usual sense because a(2n+1) = 0, while a(2n) -> oo when n -> oo. Now, the successive arithmetic means c(n) of the first n terms of the sequence are 0/1, 0/2, 2/3, 2/4, 6/5, 6/6, 12/7, 12/8, 20/9, 20/10, ... so c(2n)= (n*(n+1))/(2*n+1) ~ n/2 and c(2n+1) = n/2, hence the Cesàro limit is infinity because c(n) -> oo as n -> oo (Arnaudiès et al.), QED.
The first few irreducible fractions c(n) are in the last row of the Example section. The differences between row 4 and last row exist only when n = 4*k+1, k>0, then respectively c(n) = 2k/2 = k/1.
This sequence consists of the oblong numbers (A002378) interlaced with the natural numbers (A001477)
Note that A033999 is a counterexample in the case of a finite Cesàro limit.
Also, the converse of the Cesàro mean theorem is true iff u(n) is monotonic.
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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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FORMULA
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EXAMPLE
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Table with the first few terms:
Indices n : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
A237420(n) : 0, 0, 2, 0, 4, 0, 6, 0, 8, 0, ...
Partial sums : 0, 0, 2, 2, 6, 6, 12, 12, 20, 20, ...
Cesàro means c(n) : 0/1, 0/2, 2/3, 1/2, 6/5, 2/2, 12/7, 3/2, 20/9, 4/2, ...
Numerator a(n) : 0, 0, 2, 1, 6, 2, 12, 3, 20, 4, ...
Denominator A141310(n) : 1, 2, 3, 2, 5, 2, 7, 2, 9, 2, ...
Irreducible Cesàro mean : 0/1, 0/2, 2/3, 1/2, 6/5, 1/1, 12/7, 3/2, 20/9, 2/1, ...
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MATHEMATICA
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m = 50; Accumulate[Table[If[OddQ[n], 0, n], {n, 0, 2*m - 1}]] * Flatten[Table[{2*n - 1, 2}, {n, 1, m}]] / Range[2*m] (* Amiram Eldar, Jun 05 2022 *)
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PROG
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(PARI) c(n) = sum(k=0, n, if (k%2, 0, k))/(n+1);
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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