A243224 -id:A243224 - cp's OEIS frontend

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

Jean-Christophe Hervé, Jun 01 2014

nonn

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.

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

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.

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

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

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.

cp's OEIS Frontend

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

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