A049994 a(n) is the number of arithmetic progressions of 4 or more positive integers, nondecreasing with sum n.

0, 0, 0, 1, 1, 1, 1, 2, 1, 3, 1, 3, 1, 3, 3, 4, 1, 4, 1, 6, 3, 4, 1, 6, 4, 4, 3, 7, 1, 9, 1, 6, 3, 5, 7, 10, 1, 5, 3, 12, 1, 10, 1, 8, 10, 6, 1, 11, 4, 12, 4, 9, 1, 11, 9, 12, 4, 7, 1, 20, 1, 7, 9, 11, 10, 13, 1, 10, 4, 21, 1, 18, 1, 8, 14, 11, 7, 14, 1, 22, 8, 9, 1, 21, 12, 9, 5, 15, 1, 29, 8

Offset

1,8

Links

Antti Karttunen, Table of n, a(n) for n = 1..12580

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.

Jon Maiga, Computer-generated formulas for A049994, Sequence Machine.

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.

Formula

G.f.: Sum_{k >= 4} x^k/(1-x^(k*(k-1)/2))/(1-x^k). [Leroy Quet from A049988] - Petros Hadjicostas, Sep 29 2019

a(n) = A049992(n) - A175676(n) = A049986(n) + A321014(n). [Two of the formulas listed by Sequence Machine, both obviously true] - Antti Karttunen, Feb 20 2023

Prog

(PARI) A049994(n) = (A049992(n)-if(n%3, 0, n/3)); \\ Antti Karttunen, Feb 20 2023

Crossrefs

Cf. A014405, A014406, A175676, A049980, A049981, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A049991, A049992, A049993, A127938, A321014.

Sequence in context: A025820 A109704 A073407 * A135732 A342241 A322584

Adjacent sequences: A049991 A049992 A049993 * A049995 A049996 A049997

Keyword

nonn

Author

Clark Kimberling

Extensions

More terms from Petros Hadjicostas, Sep 29 2019

Status

approved