A127938 Number of arithmetic progressions of 2 or more nonnegative integers, strictly increasing with sum n.
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
Offset: 1
Keywords
Examples
a(10) = 7 because there are five 2-element arithmetic progressions that sum to 10, as well as 1+2+3+4 and 0+1+2+3+4.
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.
- 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.
- Wikipedia, Arithmetic progression.
Crossrefs
Programs
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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
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(k-1)}/(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 nonnegative integers, strictly increasing with sum n.
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