

A127938


Number of arithmetic progressions of 2 or more nonnegative integers, strictly increasing with sum n.


14



1, 1, 3, 2, 3, 6, 4, 4, 8, 7, 6, 11, 7, 8, 15, 9, 9, 17, 10, 13, 20, 13, 12, 22, 15, 15, 24, 18, 15, 32, 16, 18, 29, 20, 22, 36, 19, 22, 34, 27, 21, 42, 22, 26, 46, 27, 24, 45, 27, 34, 45, 31, 27, 52, 35, 35, 50, 34, 30, 64, 31, 36, 59, 38, 40, 65, 34, 40, 60, 51, 36, 71, 37, 43
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OFFSET

1,3


COMMENTS

From Petros Hadjicostas, Sep 28 2019: (Start)
We want to find the number of pairs of integers (b, w) such that b >= 0 and w >= 1 and there is an integer m >= 1 so that m*b + (1/2)*m*(m1)*w = n.
If we insist that b > 0, we get A049982 (= number of arithmetic progressions of 2 or more positive integers, strictly increasing with sum n). The number of integers m >= 1 such that (1/2)*m*(m1)*w = n equals A007862(n) (= number of triangular numbers that divide n).
Thus, to get a(n), we add A049982(n) to A007862(n).
(End)


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), 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.
Wikipedia, Arithmetic progression.


FORMULA

G.f.: x/(x^3  x  x^2 + 1) + x^3/(x^6  x^3  x^3 + 1) + x^6/(x^10  x^6  x^4 + 1) + ... = Sum_{k >= 2} x^{t(k1)}/(x^{t(k)}  x^{t(k1)}  x^k + 1), where t(k) = A000217(k) is the kth triangular number. Term k of this generating function generates the number of arithmetic progressions of k nonnegative integers, strictly increasing with sum n.
a(n) = A049982(n) + A007862(n).  Petros Hadjicostas, Sep 28 2019


EXAMPLE

a(10) = 7 because there are five 2element arithmetic progressions that sum to 10, as well as 1+2+3+4 and 0+1+2+3+4.


PROG

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


CROSSREFS

Cf. A000217, A007862, A014405, A014406, A049980, A049981, A049982, A049983, A049986, A049987.
Sequence in context: A130459 A323621 A144216 * A131990 A033771 A033795
Adjacent sequences: A127935 A127936 A127937 * A127939 A127940 A127941


KEYWORD

nonn


AUTHOR

Graeme McRae, Feb 08 2007


STATUS

approved



