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A239141 Number of strict partitions of n having standard deviation σ <= 1. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Regarding standard deviation, see Comments at A238616.

LINKS

Table of n, a(n) for n=1..86.

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., 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 *)

CROSSREFS

Cf. A239140, A239142, A239143, A000009, A238616.

Sequence in context: A318955 A173883 A022922 * A195352 A103507 A219252

Adjacent sequences:  A239138 A239139 A239140 * A239142 A239143 A239144

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Mar 11 2014

STATUS

approved

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Last modified October 22 04:25 EDT 2019. Contains 328315 sequences. (Running on oeis4.)