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
Examples
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.
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).
Programs
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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 *) -
PARI
A083039(n) = (1+!(n%2)+!(n%3)); A239140(n) = if(n<=3,1+(3==n),A083039(n-3)); \\ Antti Karttunen, May 24 2021
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
Extensions
A-number in the first formula corrected by Antti Karttunen, May 24 2021
Comments