A049983 a(n) is the number of arithmetic progressions of 2 or more positive integers, strictly increasing with sum <= n.

0, 0, 1, 2, 4, 7, 10, 13, 19, 24, 29, 37, 43, 50, 62, 70, 78, 92, 101, 112, 129, 141, 152, 171, 185, 199, 221, 237, 251, 278, 293, 310, 337, 356, 377, 409, 427, 448, 480, 505, 525, 563, 584, 609, 651, 677, 700, 742, 768, 800, 843, 873, 899, 948, 981, 1014, 1062, 1095, 1124, 1183, 1213, 1248, 1304, 1341, 1380

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Examples

			a(7) = 10 because we have the following arithmetic progressions of two or more positive integers, strictly increasing with sum <= n = 7: 1+2, 1+3, 1+4, 1+5, 1+6, 2+3, 2+4, 2+5, 3+4, and 1+2+3. - _Petros Hadjicostas_, Sep 27 2019
		

Links

• 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.

Crossrefs

Cf. A014405, A014406, A049980, A049981, A049982, A049986, A049987, A068322, A127938, A175342.

Formula

From Petros Hadjicostas, Sep 29 2019: (Start)

a(n) = Sum_{k = 1..n} A049982(k) = -n + Sum_{k = 1..n} A049980(k) = -n + A049981(k).

G.f.: (g.f. of A049982)/(1-x). (End)

Extensions

More terms from Petros Hadjicostas, Sep 27 2019