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A290889 Number of partitions of the set of odd numbers {1, 3, ..., 2*n-1} into two subsets such that the absolute difference of the sums of the two subsets is minimized.

%I A290889 #35 Nov 14 2020 08:35:09
%S A290889 1,1,1,1,2,1,5,4,13,10,38,34,118,103,380,346,1262,1153,4277,3965,
%T A290889 14745,13746,51541,48396,182182,171835,650095,615966,2338706,2223755,
%U A290889 8472697,8082457,30884150,29543309,113189168,108545916,416839177,400623807,1541726967
%N A290889 Number of partitions of the set of odd numbers {1, 3, ..., 2*n-1} into two subsets such that the absolute difference of the sums of the two subsets is minimized.
%C A290889 Partitioning in equal sums is only possible for n = 4*k-1, k > 1, and the number of such partitions is given by A156700. For the set {1,3} and the other values of n, i.e., for the sets {1,3,5}, {1,3,5,7,9}, {1,3,5,7,9,11,13}, one can use the criterion to split the sets "as well as possible" by choosing those partitions for which the absolute value of the difference of the respective sums of the subset members achieves its minimum.
%H A290889 Alois P. Heinz, <a href="/A290889/b290889.txt">Table of n, a(n) for n = 1..1000</a>
%F A290889 a(n) ~ (3 - (-1)^n) * sqrt(3) * 2^(n - 5/2) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Sep 18 2017
%e A290889 a(1) = 1: {}U{1} with difference 1.
%e A290889 a(2) = 1: {1}U{3} with difference 2.
%e A290889 a(3) = 1: {1,3}U{5} with difference 1.
%e A290889 a(4) = 1 = A156700(2): {1,7}U{3,5} with difference 0.
%e A290889 a(5) = 2: {1,3,9}U{5,7} and {1,5,7}U{3,9} with |difference|=1.
%e A290889 a(6) = 1 = A156700(3): {1,3,5,9}U{7,11} with difference 0.
%e A290889 a(7) = 5: {1,3,5,7,9}U{11,13}, {1,3,9,11}U{5,7,13}, {1,5,7,11}U{3,9,13},
%e A290889           {1,11,13}U{3,5,7,9}, {1,3,7,13}U{5,9,11} with |difference|=1.
%p A290889 b:= proc(n, i) option remember; `if`(n>i^2, 0,
%p A290889       `if`(n=i^2, 1, b(abs(n-2*i+1), i-1)+b(n+2*i-1, i-1)))
%p A290889     end:
%p A290889 a:= n-> `if`(n<5, 1, (t-> b(t, n)/(2-t))(irem(n, 2))):
%p A290889 seq(a(n), n=1..50);  # _Alois P. Heinz_, Aug 14 2017
%t A290889 b[n_, i_] := b[n, i] = If[n > i^2, 0, If[n == i^2, 1, b[Abs[n - 2i + 1], i - 1] + b[n + 2i - 1, i - 1]]];
%t A290889 a[n_] := If[n < 5, 1, b[#, n]/(2-#)&[Mod[n, 2]]];
%t A290889 Array[a, 50] (* _Jean-François Alcover_, Nov 14 2020, after _Alois P. Heinz_ *)
%Y A290889 Cf. A005408, A058377, A069918, A156700.
%K A290889 nonn
%O A290889 1,5
%A A290889 _Hugo Pfoertner_, Aug 13 2017