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A014405
Number of arithmetic progressions of 3 or more positive integers, strictly increasing with sum n.
18
0, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 3, 0, 1, 5, 1, 0, 6, 0, 2, 7, 2, 0, 8, 2, 2, 9, 3, 0, 13, 0, 2, 11, 3, 4, 15, 0, 3, 13, 6, 0, 18, 0, 4, 20, 4, 0, 19, 2, 8, 18, 5, 0, 23, 6, 6, 20, 5, 0, 30, 0, 5, 25, 6, 7, 29, 0, 6, 24, 15, 0, 32, 0, 6, 34, 7, 4, 34, 0, 14, 31, 7, 0, 39, 9, 7, 31, 9, 0, 49, 5, 9, 33, 8, 10, 42, 0, 12
OFFSET
1,9
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..12580 (first 1000 terms from Fausto A. C. Cariboni)
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.
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.
FORMULA
G.f.: Sum_{k >= 3} x^t(k)/(x^t(k) - x^t(k-1) - x^k + 1) = Sum_{k >= 3} 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
a(n) = A049992(n) - A023645(n). - Antti Karttunen, Feb 20 2023
EXAMPLE
E.g., 15 = 1+2+3+4+5 = 1+5+9 = 2+5+8 = 3+5+7 = 4+5+6.
PROG
(PARI) a(n)= t=0; st=0; forstep(s=(n-3)\3, 1, -1, st++; for(c=1, st, m=3; w=m*(s+c); while(w<n, w=w+s+m*c; m++); if(w==n, t++))); t \\ Rick L. Shepherd, Aug 30 2006
KEYWORD
nonn
STATUS
approved