A068324 Number of nondecreasing arithmetic progressions of positive odd integers with sum n.

1, 1, 2, 2, 2, 3, 2, 3, 4, 4, 2, 5, 2, 5, 6, 6, 2, 7, 2, 7, 7, 7, 2, 9, 4, 8, 8, 10, 2, 11, 2, 10, 9, 10, 5, 14, 2, 11, 10, 14, 2, 14, 2, 14, 15, 13, 2, 17, 4, 15, 12, 17, 2, 17, 6, 18, 13, 16, 2, 22, 2, 17, 17, 21, 7, 21, 2, 21, 15, 21, 2, 25, 2, 20, 21, 24, 5, 24, 2, 26, 19, 22, 2, 29, 8

Offset: 1

Author

Naohiro Nomoto, Feb 27 2002

Keywords

• easy
• nonn

Examples

			From _Petros Hadjicostas_, Sep 29 2019: (Start)
a(6) = 3 because we have the following nondecreasing arithmetic progressions of positive odd integers with sum n=6: 1+5, 3+3, and 1+1+1+1+1+1.
a(7) = 2 because we have the following nondecreasing arithmetic progressions of positive odd integers with sum n=7: 7 and 1+1+1+1+1+1+1.
a(8) = 3 because we have the following nondecreasing arithmetic progressions of positive odd integers with sum n=8: 1+7, 3+5, and 1+1+1+1+1+1+1+1.
(End)
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions, Rostok. Math. Kolloq. 64 (2009), 11-16.
• Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions with an odd common difference, Integers 9(1) (2009), 77-81.
• Graeme McRae, Counting arithmetic sequences whose sum is n.
• Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
• Augustine O. Munagi, Combinatorics of integer partitions in arithmetic progression, Integers 10(1) (2010), 73-82.
• Augustine O. Munagi and Temba Shonhiwa, On the partitions of a number into arithmetic progressions, Journal of Integer Sequences 11 (2008), Article 08.5.4.
• A. N. Pacheco Pulido, Extensiones lineales de un poset y composiciones de números multipartitos, Maestría thesis, Universidad Nacional de Colombia, 2012.
• Wikipedia, Arithmetic progression.
• Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs

Cf. A049980, A049981, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A068322, A068323, A127938, A175327, A325328, A325407, A325545, A325546, A325547, A325548.

Formula

From Petros Hadjicostas, Oct 01 2019: (Start)

a(n) = A068322(n) + A001227(n) - (1/2) * (1 - (-1)^n).

G.f.: x/(1 - x^2) + Sum_{m >= 2} x^m/((1 - x^(2*m)) * (1 - x^(m*(m-1)))).

(End)

Extensions

Extended and edited by John W. Layman, Mar 15 2002