A243223 Number of partitions of n into positive summands in arithmetic progression with common difference 3.

0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 2, 0, 1, 2, 1, 0, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 0, 3, 1, 1, 1, 1, 2, 3, 0, 1, 2, 2, 0, 3, 1, 1, 2, 1, 1, 3, 0, 2, 2, 1, 0, 3, 3, 1, 1, 1, 1, 4, 0, 2, 2, 1, 1, 3, 1, 1, 2, 2, 1, 3, 0, 1, 3, 2, 1, 3, 1, 2, 1, 1

Offset

1,15

Comments

This sequence gives the number of ways to write n as n = a + a+3 + ... + a+3r = (r+1)(2a+3r)/2, with a and r integers > 0.

Links

Jean-Christophe Hervé, Table of n, a(n) for n = 1..10045

J. W. Andrushkiw, R. I. Andrushkiw and C. E. Corzatt, Representations of Positive Integers as Sums of Arithmetic Progressions, Mathematics Magazine, Vol. 49, No. 5 (Nov., 1976), pp. 245-248.

M. A. Nyblom and C. Evans, On the enumeration of partitions with summands in arithmetic progression, Australasian Journal of Combinatorics, Vol. 28 (2003), pp. 149-159.

Formula

a(n) = d1(n) - 1 - f(n) with d1(n) = number of odd divisors of n (A001227) and f(n) = the number of those odd divisors d of n such that d > 1 and d(1+d/3)/2 <= n <= 3d(d-1)/2. f(n) is in A243224.

Example

a(15) = 2 because 15 = 6 + 9 = 2 + 5 + 8.

Crossrefs

Cf. A072670 (same with common differences = 2).

A243225 gives the integers n that are not such sums for which a(n) = 0.

Sequence in context: A088434 A205745 A333781 * A034178 A317531 A074169

Adjacent sequences: A243220 A243221 A243222 * A243224 A243225 A243226

Keyword

nonn

Author

Jean-Christophe Hervé, Jun 01 2014

Status

approved