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

Offset: 0

Author

Bernard Schott, May 22 2022

Keywords

• nonn
• easy

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.

Examples

			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, ...
		

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

• 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).

Crossrefs

Cf. A001477, A002378, A033999, A141310 (denominators), A237420.

Programs

Mathematica

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 *)

PARI

c(n) = sum(k=0, n, if (k%2, 0, k))/(n+1);
f(n) = if(n%2, 2, 1+n); \\ A141310
a(n) = c(n)*f(n); \\ Michel Marcus, Jun 06 2022

Formula

a(n) = (A141310(n)/(n+1)) * Sum_{k=0..n} A237420(k).

For n >= 0, a(2n) = n*(n+1) = A002378(n), a(2n+1) = n = A001477(n).

G.f.: x^2*(2 + x - x^3)/(1 - x^2)^3. - Stefano Spezia, May 23 2022