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%I #61 Sep 30 2019 01:28:25
%S 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,
%T 14,27,15,17,27,19,21,32,18,21,32,25,20,38,21,25,42,26,23,42,26,32,43,
%U 30,26,49,33,33,48,33,29,59,30,35,56,37,39,61,33,39,58,49,35,67,36,42
%N Number of arithmetic progressions of 2 or more positive integers, strictly increasing with sum n.
%H Andrew Howroyd, <a href="/A049982/b049982.txt">Table of n, a(n) for n = 1..10000</a>
%H Sadek Bouroubi and Nesrine Benyahia Tani, <a href="http://ftp.math.uni-rostock.de/pub/romako/heft64/bou64.pdf">Integer partitions into arithmetic progressions</a>, Rostok. Math. Kolloq. 64 (2009), 11-16.
%H Graeme McRae, <a href="https://web.archive.org/web/20081122034835/http://2000clicks.com/MathHelp/BasicSequenceA049982.htm">Counting arithmetic sequences whose sum is n</a>.
%H Graeme McRae, <a href="/A049988/a049988.pdf">Counting arithmetic sequences whose sum is n</a> [Cached copy]
%H Augustine O. Munagi and Temba Shonhiwa, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Shonhiwa/shonhiwa13.html">On the partitions of a number into arithmetic progressions</a>, Journal of Integer Sequences 11 (2008), Article 08.5.4.
%H A. N. Pacheco Pulido, <a href="http://www.bdigital.unal.edu.co/7753/">Extensiones lineales de un poset y composiciones de números multipartitos</a>, Maestría thesis, Universidad Nacional de Colombia, 2012.
%F 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
%F From _Petros Hadjicostas_, Sep 27 2019: (Start)
%F a(n) = A049980(n) - 1 = A049988(n) - A000005(n).
%F a(n) = A049981(n) - A049981(n-1) - 1 for n >= 2.
%F 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))).
%F (End)
%o (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
%Y Cf. A000005, A000217, A014405, A014406, A049980, A049981, A049983, A049986, A049987, A049988, A068322, A127938, A175342.
%K nonn
%O 1,5
%A _Clark Kimberling_
%E More terms from _Petros Hadjicostas_, Sep 28 2019
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