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

%I A239140 #17 May 24 2021 23:31:54
%S A239140 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,
%T A239140 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,
%U A239140 3,1,2,2,2,1,3,1,2,2,2,1,3,1,2,2,2,1
%N A239140 Number of strict partitions of n having standard deviation σ < 1.
%C A239140 Regarding standard deviation, see Comments at A238616.
%H A239140 Antti Karttunen, <a href="/A239140/b239140.txt">Table of n, a(n) for n = 1..10005</a>
%H A239140 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-1, 0, 1, 1).
%F A239140 a(n + 3) = A083039(n) for n >= 1 (periodic with period 6); a(n) + A239143(n) = A000009(n) for n >=1.
%F A239140 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
%e A239140 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.
%t A239140 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]]
%t A239140 Table[Count[g[n], p_ /; s[p] < 1], {n, z}]   (* A239140 *)
%t A239140 Table[Count[g[n], p_ /; s[p] <= 1], {n, z}]  (* A239141 *)
%t A239140 Table[Count[g[n], p_ /; s[p] == 1], {n, z}]  (* periodic 01 *)
%t A239140 Table[Count[g[n], p_ /; s[p] > 1], {n, z}]   (* A239142 *)
%t A239140 Table[Count[g[n], p_ /; s[p] >= 1], {n, z}]  (* A239143 *)
%t A239140 t[n_] := t[n] = N[Table[s[g[n][[k]]], {k, 1, PartitionsQ[n]}]]
%t A239140 ListPlot[Sort[t[30]]] (*plot of st.dev's of strict partitions of 30*)
%t A239140 (* _Peter J. C. Moses_, Mar 03 2014 *)
%t A239140 Join[{1, 1, 2},LinearRecurrence[{-1, 0, 1, 1},{1, 2, 2, 2},83]] (* _Ray Chandler_, Aug 25 2015 *)
%o A239140 (PARI)
%o A239140 A083039(n) = (1+!(n%2)+!(n%3));
%o A239140 A239140(n) = if(n<=3,1+(3==n),A083039(n-3)); \\ _Antti Karttunen_, May 24 2021
%Y A239140 Cf. A000009, A083039, A238616, A239141, A239142, A239143.
%Y A239140 Column k=0 of A239228.
%K A239140 nonn,easy
%O A239140 1,3
%A A239140 _Clark Kimberling_, Mar 11 2014
%E A239140 A-number in the first formula corrected by _Antti Karttunen_, May 24 2021