A049980 a(n) is the number of arithmetic progressions of positive integers, strictly increasing with sum n.
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
Offset: 1
Keywords
Examples
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
Links
- Fausto A. C. Cariboni, 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.
- 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.
Crossrefs
Formula
Conjecture: a(n) = 1 + 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))). - Petros Hadjicostas, Sep 27 2019
G.f.: x/(1-x) + Sum_{k >= 2} x^t(k)/(x^t(k) - x^t(k-1) - x^k + 1) = x/(1-x) + Sum_{k >= 2} x^t(k)/((1 - x^k) * (1 - x^t(k-1))), where t(k) = k*(k+1)/2 = A000217(k) is the k-th triangular number [Graeme McRae]. - Petros Hadjicostas, Sep 29 2019
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