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

1,2

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.

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).

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

Example

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, ...

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 *)

Prog

(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

Crossrefs

Cf. A005408, A114112, A130883, A141310.

Sequence in context: A324821 A345401 A059912 * A115765 A282598 A269369

Adjacent sequences: A354005 A354006 A354007 * A354009 A354010 A354011

Keyword

nonn,easy,frac

Author

Bernard Schott, May 13 2022

Status

approved