%I
%S 1,1,1,1,2,1,2,1,2,2,3,1,3,2,2,1
%N a(n) is the number of halving partitions of n (see Comments for definition).
%C Let S(m) = {f(m/2)), c(m/2) : m >= 1}, where f = floor and c = ceiling. A halving partition of n is a partition p(1) + p(2) + ... + p(k) of n such that p(1) is in S(n) and p(i) is in S(p(i1)) for i = 1, 2, ..., k. The basic idea is that each term after the first is about half of the preceding term.
%e a(9) counts these 2 partitions:
%e c(9/2) + f(5/2) + f(2/2} + c(1/2) = 5 + 3 + 1;
%e f(9/2) + f(5/2) + f(2/2) + c(1/2) = 4 + 2 + 1 + 1.
%e a(13) counts these:
%e c(13/2) + c(7/2) + f(4/2) = 7 + 4 + 2;
%e c(13/2) + f(7/2) + c(3/2) + f(2/2) = 7 + 3 + 2 + 1;
%e f(13/2) + f(6/2) + c(3/2) + f(2/2) + c(1/2) = 6 + 3 + 2 + 1 + 1.
%K nonn,more
%O 1,5
%A _Clark Kimberling_, Dec 26 2021
