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A349552 a(n) is the number of halving partitions of n (see Comments for definition).

%I A349552 #38 Jan 07 2025 10:15:07
%S A349552 1,1,1,2,1,3,2,3,1,3,3,5,2,5,3,4,1,4,3,6,3,7,5,7,2,6,5,8,3,7,4,5,1,4,
%T A349552 4,7,3,9,6,9,3,9,7,12,5,11,7,9,2,8,6,11,5,12,8,11,3,9,7,11,4,9,5,6,1,
%U A349552 4,4,8,4,10,7,11,3,11,9,15,6,15,9,12,3,10,9,16,7,18,12,17,5,15,11,18,7,15,9,11,2,8,8,14,6
%N A349552 a(n) is the number of halving partitions of n (see Comments for definition).
%C A349552 For m >= 1, let S(m) = { f(m/2), c(m/2) }, 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(i-1)) for i = 2, 3, ..., k. The basic idea is that each term after the first is about half of the preceding term.
%H A349552 Alois P. Heinz, <a href="/A349552/b349552.txt">Table of n, a(n) for n = 0..32768</a>
%F A349552 From _Alois P. Heinz_, Sep 30 2024: (Start)
%F A349552 a(A000079(n)) = 1.
%F A349552 a(A000225(n)) = A028310(n). (End)
%e A349552 a(9) = 3 counts these partitions:
%e A349552   c(9/2) + f(5/2) + (2/2) + c(1/2) = 5 + 2 + 1 + 1;
%e A349552   c(9/2) + c(5/2) + f(3/2) = 5 + 3 + 1;
%e A349552   f(9/2) + (4/2) + (2/2) + c(1/2) = 4 + 2 + 1 + 1.
%e A349552 a(13) = 5 counts these partitions:
%e A349552   c(13/2) + c(7/2) + (4/2) = 7 + 4 + 2;
%e A349552   c(13/2) + f(7/2) + c(3/2) + (2/2) = 7 + 3 + 2 + 1;
%e A349552   c(13/2) + f(7/2) + f(3/2) + (2/2) + c(1/2) = 7 + 3 + 1 + 1 + 1;
%e A349552   f(13/2) + (6/2) + c(3/2) + (2/2) + c(1/2) = 6 + 3 + 2 + 1 + 1;
%e A349552   f(13/2) + (6/2) + f(3/2) + (2/2) + c(1/2) + c(1/2) = 6 + 3 + 1 + 1 + 1 + 1.
%o A349552 (PARI) { a349552(n,p=n) = if(n==0,1,if(n<0||p==0,0,if(p%2,a(n-p\2-1,p\2+1))+a(n-p\2,p\2))); } \\ _Max Alekseyev_, Sep 30 2024
%Y A349552 Cf. A000079, A000225, A028310, A349553.
%K A349552 nonn,look
%O A349552 0,4
%A A349552 _Clark Kimberling_, Dec 26 2021
%E A349552 Corrected and extended by _Max Alekseyev_, Sep 30 2024