%I
%S 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,
%T 15,28,16,18,28,20,22,33,19,22,33,26,21,39,22,26,43,27,24,43,27,33,44,
%U 31,27,50,34,34,49,34,30,60,31,36,57,38,40
%N a(n) is the number of arithmetic progressions of positive integers, strictly increasing with sum n.
%C We need to find the number of pairs of positive integers (b, w) so that there is a positive integer m such that m*b + m*(m1)*w/2 = n.  _Petros Hadjicostas_, Sep 27 2019
%H Sadek Bouroubi and Nesrine Benyahia Tani, <a href="http://ftp.math.unirostock.de/pub/romako/heft64/bou64.pdf">Integer partitions into arithmetic progressions</a>, Rostok. Math. Kolloq. 64 (2009), 1116.
%H Sadek Bouroubi and Nesrine Benyahia Tani, <a href="https://www.emis.de/journals/INTEGERS/papers/j7/j7.Abstract.html">Integer partitions into arithmetic progressions with an odd common difference</a>, Integers 9(1) (2009), 7781.
%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, <a href="https://www.emis.de/journals/INTEGERS/papers/k7/k7.Abstract.html">Combinatorics of integer partitions in arithmetic progression</a>, Integers 10(1) (2010), 7382.
%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 Conjecture: a(n) = 1 + Sum_{mn, m odd > 1} floor(2 * (n  m)/(m* (m  1))) + Sum_{mn} floor((n  m * (5  (1)^(n/m))/2 + m^2 * (1  (1)^(n/m)))/(2*m * (2*m  1))).  _Petros Hadjicostas_, Sep 27 2019
%F G.f.: x/(1x) + Sum_{k >= 2} x^t(k)/(x^t(k)  x^t(k1)  x^k + 1) = x/(1x) + Sum_{k >= 2} x^t(k)/((1  x^k) * (1  x^t(k1))), where t(k) = k*(k+1)/2 = A000217(k) is the kth triangular number [_Graeme McRae_].  _Petros Hadjicostas_, Sep 29 2019
%e 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
%Y Cf. A000217, A014405, A014406, A049981, A049982, A049983, A049986, A049987, A068322, A068323, A068324, A127938, A175342.
%K nonn
%O 1,3
%A _Clark Kimberling_
