A014405 Number of arithmetic progressions of 3 or more positive integers, strictly increasing with sum n.

0, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 3, 0, 1, 5, 1, 0, 6, 0, 2, 7, 2, 0, 8, 2, 2, 9, 3, 0, 13, 0, 2, 11, 3, 4, 15, 0, 3, 13, 6, 0, 18, 0, 4, 20, 4, 0, 19, 2, 8, 18, 5, 0, 23, 6, 6, 20, 5, 0, 30, 0, 5, 25, 6, 7, 29, 0, 6, 24, 15, 0, 32, 0, 6, 34, 7, 4, 34, 0, 14, 31, 7, 0, 39, 9, 7, 31, 9, 0, 49, 5, 9, 33, 8, 10, 42, 0, 12

Offset

1,9

Links

Antti Karttunen, Table of n, a(n) for n = 1..12580 (first 1000 terms from Fausto A. C. Cariboni)

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.

Formula

G.f.: Sum_{k >= 3} x^t(k)/(x^t(k) - x^t(k-1) - x^k + 1) = Sum_{k >= 3} 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) = A049992(n) - A023645(n). - Antti Karttunen, Feb 20 2023

Example

E.g., 15 = 1+2+3+4+5 = 1+5+9 = 2+5+8 = 3+5+7 = 4+5+6.

Prog

(PARI) a(n)= t=0; st=0; forstep(s=(n-3)\3, 1, -1, st++; for(c=1, st, m=3; w=m*(s+c); while(w<n, w=w+s+m*c; m++); if(w==n, t++))); t // Rick L. Shepherd, Aug 30 2006

Crossrefs

Cf. A000217, A007862, A014406, A014407, A023645, A047966, A049982, A049983, A049986, A049987, A049992, A129654, A240026, A240027, A307824, A320466, A325325, A325328.

Sequence in context: A364106 A364033 A050464 * A368464 A143153 A127448

Adjacent sequences: A014402 A014403 A014404 * A014406 A014407 A014408

Keyword

nonn

Author

Clark Kimberling

Status

approved