

A049980


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


16



1, 1, 2, 2, 3, 4, 4, 4, 7, 6, 6, 9, 7, 8, 13, 9, 9, 15, 10, 12, 18, 13, 12, 20, 15, 15, 23, 17, 15, 28, 16, 18, 28, 20, 22, 33, 19, 22, 33, 26, 21, 39, 22, 26, 43, 27, 24, 43, 27, 33, 44, 31, 27, 50, 34, 34, 49, 34, 30, 60, 31, 36, 57, 38, 40
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OFFSET

1,3


COMMENTS

We need to find the number of pairs of positive integers (b, w) so that there is a positive integer m such that m*b + m*(m1)*w/2 = n.  Petros Hadjicostas, Sep 27 2019


LINKS

Table of n, a(n) for n=1..65.
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions, Rostok. Math. Kolloq. 64 (2009), 1116.
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions with an odd common difference, Integers 9(1) (2009), 7781.
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), 7382.
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.


FORMULA

Conjecture: a(n) = 1 + Sum_{mn, m odd > 1} floor(2 * (n  m)/(m* (m  1))) + Sum_{mn} floor((n  m * (5  (1)^(n/m))/2 + m^2 * (1  (1)^(n/m)))/(2*m * (2*m  1))).  Petros Hadjicostas, Sep 27 2019
G.f.: x/(1x) + Sum_{k >= 2} x^t(k)/(x^t(k)  x^t(k1)  x^k + 1) = x/(1x) + Sum_{k >= 2} x^t(k)/((1  x^k) * (1  x^t(k1))), where t(k) = k*(k+1)/2 = A000217(k) is the kth triangular number [Graeme McRae].  Petros Hadjicostas, Sep 29 2019


EXAMPLE

a(6) = 4 because we have the following strictly increasing arithmetic progressions of positive integers adding up to n = 6: 6, 1+5, 2+4, and 1+2+3.  Petros Hadjicostas, Sep 27 2019


CROSSREFS

Cf. A000217, A014405, A014406, A049981, A049982, A049983, A049986, A049987, A068322, A068323, A068324, A127938, A175342.
Sequence in context: A130128 A210556 A208914 * A209698 A141525 A209764
Adjacent sequences: A049977 A049978 A049979 * A049981 A049982 A049983


KEYWORD

nonn


AUTHOR

Clark Kimberling


STATUS

approved



