A354280 - OEIS

A354280

a(n) is the numerator of Cesàro means sequence c(n) of A237420 when the denominator is A141310(n).

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

0,3

COMMENTS

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.

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

Table of n, a(n) for n=0..68.

ProofWiki, Cesàro mean.

The MacTutor History of Mathematics archive, Ernesto Cesàro.

Wikipedia, Ernesto Cesàro.

Wikipédia, Lemme de Cesàro (in French).

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).