A049986 a(n) is the number of arithmetic progressions of 4 or more positive integers, strictly increasing with sum n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 2, 1, 2, 0, 1, 2, 2, 1, 3, 0, 4, 0, 2, 1, 3, 4, 4, 0, 3, 1, 6, 0, 5, 0, 4, 6, 4, 0, 4, 2, 8, 2, 5, 0, 6, 6, 6, 2, 5, 0, 11, 0, 5, 5, 6, 7, 8, 0, 6, 2, 15, 0, 9, 0, 6, 10, 7, 4, 9, 0, 14, 5, 7, 0, 12, 9, 7, 3

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

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 A049986, 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.

Crossrefs

Cf. A014405, A014406, A049980, A049981, A049982, A049983, A049987 (partial sums), A049988, A049989, A049990, A049991, A049994, A127938, A321014.

Programs

PARI

\\ Needs also code from A014405 and A023645:
A049994(n) = (A014405(n) + A023645(n) - if(n%3, 0, n/3));
A321014(n) = (numdiv(n) - 3 + !!(n%2) + !!(n%3)); \\ From A321014.
A049986(n) = (A049994(n)-A321014(n)); \\ Antti Karttunen, Feb 20 2023

Formula

G.f.: Sum_{k >= 4} x^t(k)/(x^t(k) - x^t(k-1) - x^k + 1) = Sum_{k >= 4} x^t(k)/((1 - x^k)*(1 - x^t(k-1))), where t(k) = k*(k+1)/2 = A000217(k) is the k-th triangular number [Graeme McRae]. - Petros Hadjicostas, Sep 29 2019

a(n) = A049994(n) - A321014(n). [Listed by Sequence Machine and obviously true] - Antti Karttunen, Feb 20 2023