A239141 - OEIS

A239141

Number of strict partitions of n having standard deviation <= 1.

1, 1, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2

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OFFSET

1,3

COMMENTS

Regarding standard deviation, see Comments at A238616.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10005

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

FORMULA

a(n) + A239142(n) = A000009(n) for n >= 1.

G.f.: -(x^5 + x^4 + x^3 + 2*x^2 + x + 1)*x / ((x-1)*(x^2 + x + 1)). - Alois P. Heinz, Mar 14 2014

EXAMPLE

The standard deviations of the strict partitions of 9 are 0.0, 3.5, 2.5, 1.5, 2.16025, 0.5, 1.63299, 0.816497, so that a(9) = 3.

MATHEMATICA

z = 30; g[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; s[t_] := s[t] = Sqrt[Sum[(t[[k]] - Mean[t])^2, {k, 1, Length[t]}]/Length[t]]

Table[Count[g[n], p_ /; s[p] < 1], {n, z}] (* A239140 *)

Table[Count[g[n], p_ /; s[p] <= 1], {n, z}] (* A239141 *)

Table[Count[g[n], p_ /; s[p] == 1], {n, z}] (* periodic 01 *)

Table[Count[g[n], p_ /; s[p] > 1], {n, z}] (* A239142 *)

Table[Count[g[n], p_ /; s[p] >= 1], {n, z}] (* A239143 *)

t[n_] := t[n] = N[Table[s[g[n][[k]]], {k, 1, PartitionsQ[n]}]]

ListPlot[Sort[t[30]]] (*plot of st.dev's of strict partitions of 30*)

(* Peter J. C. Moses, Mar 03 2014 *)

Join[{1, 1, 2}, LinearRecurrence[{0, 0, 1}, {2, 2, 3}, 83]] (* Ray Chandler, Aug 25 2015 *)

PROG

(PARI) A239141(n) = (1+(n>3)+!(n%3)); \\ Antti Karttunen, May 24 2021

CROSSREFS