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

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, 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, 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

Offset

0,4

Comments

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.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..32768

Formula

From Alois P. Heinz, Sep 30 2024: (Start)

a(A000079(n)) = 1.

a(A000225(n)) = A028310(n). (End)

Example

a(9) = 3 counts these partitions:
  c(9/2) + f(5/2) + (2/2) + c(1/2) = 5 + 2 + 1 + 1;
  c(9/2) + c(5/2) + f(3/2) = 5 + 3 + 1;
  f(9/2) + (4/2) + (2/2) + c(1/2) = 4 + 2 + 1 + 1.
a(13) = 5 counts these partitions:
  c(13/2) + c(7/2) + (4/2) = 7 + 4 + 2;
  c(13/2) + f(7/2) + c(3/2) + (2/2) = 7 + 3 + 2 + 1;
  c(13/2) + f(7/2) + f(3/2) + (2/2) + c(1/2) = 7 + 3 + 1 + 1 + 1;
  f(13/2) + (6/2) + c(3/2) + (2/2) + c(1/2) = 6 + 3 + 2 + 1 + 1;
  f(13/2) + (6/2) + f(3/2) + (2/2) + c(1/2) + c(1/2) = 6 + 3 + 1 + 1 + 1 + 1.

Prog

(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

Crossrefs

Cf. A000079, A000225, A028310, A349553.

Sequence in context: A151682 A318928 A159918 * A278573 A108663 A379631

Adjacent sequences: A349549 A349550 A349551 * A349553 A349554 A349555

Keyword

nonn,look

Author

Clark Kimberling, Dec 26 2021

Extensions

Corrected and extended by Max Alekseyev, Sep 30 2024

Status

approved