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A049982
Number of arithmetic progressions of 2 or more positive integers, strictly increasing with sum n.
21
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
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions, Rostok. Math. Kolloq. 64 (2009), 11-16.
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
KEYWORD
nonn
EXTENSIONS
More terms from Petros Hadjicostas, Sep 28 2019
STATUS
approved