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A014406 Number of strictly increasing arithmetic progressions of positive integers with at least 3 terms and sum <= n. 14
0, 0, 0, 0, 0, 1, 1, 1, 3, 4, 4, 7, 7, 8, 13, 14, 14, 20, 20, 22, 29, 31, 31, 39, 41, 43, 52, 55, 55, 68, 68, 70, 81, 84, 88, 103, 103, 106, 119, 125, 125, 143, 143, 147, 167, 171, 171, 190, 192, 200, 218, 223, 223, 246, 252, 258, 278, 283, 283, 313, 313, 318, 343, 349, 356, 385, 385 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,9

LINKS

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..1000

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

a(n) = Sum_{k=1..n} A014405(k). - Sean A. Irvine, Oct 22 2018

G.f.: (g.f. of A014405)/(1-x). - Petros Hadjicostas, Sep 29 2019

EXAMPLE

From Petros Hadjicostas, Sep 29 2019: (Start)

a(8) = 1 because we have only the following strictly increasing arithmetic progression of positive integers with at least 3 terms and sum <= 8: 1+2+3.

a(9) = 3 because we have the following strictly increasing arithmetic progressions of positive integers with at least 3 terms and sum <= 9: 1+2+3, 1+3+5, and 2+3+4.

a(10) = 4 because we have the following strictly increasing arithmetic progressions of positive integers with at least 3 terms and sum <= 10: 1+2+3, 1+3+5, 2+3+4, and 1+2+3+4.

(End)

CROSSREFS

Cf. A007862, A014405, A047966, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A049991, A129654, A240026, A240027, A307824, A320466, A325325, A325328.

Sequence in context: A084138 A127141 A272668 * A154426 A231219 A231343

Adjacent sequences:  A014403 A014404 A014405 * A014407 A014408 A014409

KEYWORD

nonn

AUTHOR

Clark Kimberling

EXTENSIONS

a(59)-a(67) corrected by Fausto A. C. Cariboni, Oct 02 2018

STATUS

approved

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Last modified December 10 20:23 EST 2019. Contains 329909 sequences. (Running on oeis4.)