A354008 Numerators of Cesàro means sequence of A114112.

1, 3, 7, 5, 16, 7, 29, 9, 46, 11, 67, 13, 92, 15, 121, 17, 154, 19, 191, 21, 232, 23, 277, 25, 326, 27, 379, 29, 436, 31, 497, 33, 562, 35, 631, 37, 704, 39, 781, 41, 862, 43, 947, 45, 1036, 47, 1129, 49, 1226, 51, 1327, 53, 1432, 55, 1541, 57, 1654, 59, 1771, 61, 1892, 63, 2017, 65

Offset: 1

Author

Bernard Schott, May 13 2022

Keywords

• nonn
• easy
• frac

Comments

This sequence lists the numerators of c(n) = (Sum_{k=1..n} A114112(k)) / n. The corresponding denominator is A141310(n-1) (see Example section).

When a sequence u(n) is increasing, then Cesàro means sequence c(n) = (u(1)+...+u(n))/n is also increasing, but the converse is false.

A114112 is such a counterexample.

Proof: A114112 is clearly not increasing; now, the successive arithmetic means c(n) of the first specific terms of the sequence are 1/1, 3/2, 7/3, 10/4, 16/5, 21/6, 29/7, ... so, if m >= 1, c(2m) = (2m+1)/2 and c(2m+1) = m+1 + 1/(2m+1), c(1) = 1. We get c(n) = a(n) / A141310(n-1) for n >= 1.

We have c(2m+1) - c(2m) = 1/(2m+1) + 1/2 > 0 and c(2m+2) - c(2m+1) = (2m-1) / (4m+2) > 0 when m >= 1; hence for m >= 1, c(2m) < c(2m+1) < c(2m+2), and also c(1) = 1 < c(2) = 3/2; QED.

Examples

			Table with the first few terms:
       Indices n        :  1,   2,   3,   4,    5,   6,    7,   8,    9,   10, ...
       A114112(n)       :  1,   2,   4,   3,    6,   5,    8,   7,   10,    9, ...
      Partial sums      :  1,   3,   7,  10,   16,  21,   29,  36,   46,   55, ...
    Cesàro means c(n)   :  1, 3/2, 7/3, 5/2, 16/5, 7/2, 29/7, 9/2, 46/9, 11/2, ...
      Numerator a(n)    :  1,   3,   7,   5,   16,   7,   29,   9,   46,   11, ...
Denominator A141310(n-1):  1,   2,   3,   2,    5,   2,    7,   2,    9,    2, ...
		

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

• Winston de Greef, Table of n, a(n) for n = 1..10000
• ProofWiki, Cesàro mean.
• 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. A005408, A114112, A130883, A141310.

Programs

Mathematica

s[1] = 1; s[2] = 2; s[n_] := If[OddQ[n], n + 1, n - 1]; m = 100; Numerator[Accumulate[Array[s, m]]/Range[m]] (* Amiram Eldar, May 15 2022 *)

PARI

f(n) = if (n<=2, n, if (n%2, n+1, n-1)); \\ A114112
a(n) = numerator(sum(k=1, n, f(k))/n); \\ Michel Marcus, May 16 2022

Python

from math import gcd
def A354008(n): return 1 if n == 1 else (k:= (m:=n//2)*(n+1) + (n+1-m)*(n-2*m))//gcd(k,n) # Chai Wah Wu, May 24 2022

Formula

a(1) = 1, then for m >= 1: a(2m+1) = A130883(m+1) and a(2m) = A005408(m) = 2m+1.

G.f.: x*(1 + 3*x + 4*x^2 - 4*x^3 - 2*x^4 + x^5 + x^6)/(1 - x^2)^3. - Stefano Spezia, May 15 2022