A049982 Number of arithmetic progressions of 2 or more positive integers, strictly increasing with sum n.

0, 0, 1, 1, 2, 3, 3, 3, 6, 5, 5, 8, 6, 7, 12, 8, 8, 14, 9, 11, 17, 12, 11, 19, 14, 14, 22, 16, 14, 27, 15, 17, 27, 19, 21, 32, 18, 21, 32, 25, 20, 38, 21, 25, 42, 26, 23, 42, 26, 32, 43, 30, 26, 49, 33, 33, 48, 33, 29, 59, 30, 35, 56, 37, 39, 61, 33, 39, 58, 49, 35, 67, 36, 42

Offset

1,5

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.

Graeme McRae, Counting arithmetic sequences whose sum is n.

Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]

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

a(n) has generating function x^3/(x^3 - x - x^2 + 1) + x^6/(x^6 - x^3 - x^3 + 1) + x^10/(x^10 - x^6 - x^4 + 1) + ... = Sum_{k >= 2} x^t(k)/(x^t(k) - x^t(k-1) - x^k + 1), where t(k) = A000217(k) is the k-th triangular number. Term k of this generating function generates the number of arithmetic progressions of k positive integers, strictly increasing with sum n. - Graeme McRae, Feb 08 2007

From Petros Hadjicostas, Sep 27 2019: (Start)

a(n) = A049980(n) - 1 = A049988(n) - A000005(n).

a(n) = A049981(n) - A049981(n-1) - 1 for n >= 2.

Conjecture: a(n) = Sum_{m|n, m odd > 1} floor(2 * (n - m)/(m* (m - 1))) + Sum_{m|n} floor((n - m * (5 - (-1)^(n/m))/2 + m^2 * (1 - (-1)^(n/m)))/(2*m * (2*m - 1))).

(End)

Prog

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

Crossrefs

Cf. A000005, A000217, A014405, A014406, A049980, A049981, A049983, A049986, A049987, A049988, A068322, A127938, A175342.

Sequence in context: A115206 A093653 A205442 * A245642 A289559 A070167

Adjacent sequences: A049979 A049980 A049981 * A049983 A049984 A049985

Keyword

nonn

Author

Clark Kimberling

Extensions

More terms from Petros Hadjicostas, Sep 28 2019

Status

approved