A239140 Number of strict partitions of n having standard deviation σ < 1.

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

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 (-1, 0, 1, 1).

Formula

a(n + 3) = A083039(n) for n >= 1 (periodic with period 6); a(n) + A239143(n) = A000009(n) for n >=1.

G.f.: -(x^6+x^5+x^4+2*x^3+3*x^2+2*x+1)*x / ((x-1)*(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[{-1, 0, 1, 1}, {1, 2, 2, 2}, 83]] (* Ray Chandler, Aug 25 2015 *)

Prog

(PARI)
A083039(n) = (1+!(n%2)+!(n%3));
A239140(n) = if(n<=3, 1+(3==n), A083039(n-3)); \\ Antti Karttunen, May 24 2021

Crossrefs

Cf. A000009, A083039, A238616, A239141, A239142, A239143.

Column k=0 of A239228.

Sequence in context: A082641 A334507 A342462 * A138553 A069016 A211270

Adjacent sequences: A239137 A239138 A239139 * A239141 A239142 A239143

Keyword

nonn,easy

Author

Clark Kimberling, Mar 11 2014

Extensions

A-number in the first formula corrected by Antti Karttunen, May 24 2021

Status

approved