A049989 a(n) is the number of arithmetic progressions of positive integers, nondecreasing with sum <= n.

1, 3, 6, 10, 14, 21, 26, 33, 42, 51, 58, 72, 80, 91, 107, 120, 130, 150, 161, 178, 199, 215, 228, 255, 272, 290, 316, 338, 354, 389, 406, 429, 460, 483, 508, 549, 569, 594, 630, 663, 685, 731, 754, 785, 833, 863, 888, 940, 969, 1007, 1054, 1090, 1118, 1175, 1212, 1253, 1305, 1342, 1373, 1444, 1476, 1515, 1577, 1621

Offset

1,2

Links

Andrew Howroyd, 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.

Formula

From Petros Hadjicostas, Sep 29 2019: (Start)

a(n) = Sum_{k = 1..n} A049988(k). [Note that the offset of A049988 is 0.]

G.f.: (-1 + g.f. of A049988)/(1-x). (End)

Prog

(PARI) seq(n)={my(w=(sqrtint(8*n+1)-1)\2+1); Vec(x/(1-x)^2 + sum(k=2, n, x^k/(1 - if(k<=w, x^(k*(k-1)/2)))/(1-x^k) + O(x*x^n))/(1-x))} \\ Andrew Howroyd, Sep 28 2019

Crossrefs

Cf. A047966, A049982, A049983, A049984, A049986, A049987, A129654, A240026, A240027, A307824, A320466, A325325, A325328.

Sequence in context: A183863 A140949 A025206 * A330258 A197056 A104619

Adjacent sequences: A049986 A049987 A049988 * A049990 A049991 A049992

Keyword

nonn

Author

Clark Kimberling

Extensions

More terms from Petros Hadjicostas, Sep 28 2019

Status

approved