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%I #35 Jan 23 2021 17:33:39
%S 1,1,2,2,2,3,2,3,4,4,2,5,2,5,6,6,2,7,2,7,7,7,2,9,4,8,8,10,2,11,2,10,9,
%T 10,5,14,2,11,10,14,2,14,2,14,15,13,2,17,4,15,12,17,2,17,6,18,13,16,2,
%U 22,2,17,17,21,7,21,2,21,15,21,2,25,2,20,21,24,5,24,2,26,19,22,2,29,8
%N Number of nondecreasing arithmetic progressions of positive odd integers with sum n.
%H Seiichi Manyama, <a href="/A068324/b068324.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 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), 77-81.
%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), 73-82.
%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.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic_progression">Arithmetic progression</a>.
%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>.
%F From _Petros Hadjicostas_, Oct 01 2019: (Start)
%F a(n) = A068322(n) + A001227(n) - (1/2) * (1 - (-1)^n).
%F G.f.: x/(1 - x^2) + Sum_{m >= 2} x^m/((1 - x^(2*m)) * (1 - x^(m*(m-1))).
%F (End)
%e From _Petros Hadjicostas_, Sep 29 2019: (Start)
%e a(6) = 3 because we have the following nondecreasing arithmetic progressions of positive odd integers with sum n=6: 1+5, 3+3, and 1+1+1+1+1+1.
%e a(7) = 2 because we have the following nondecreasing arithmetic progressions of positive odd integers with sum n=7: 7 and 1+1+1+1+1+1+1.
%e a(8) = 3 because we have the following nondecreasing arithmetic progressions of positive odd integers with sum n=8: 1+7, 3+5, and 1+1+1+1+1+1+1+1.
%e (End)
%Y Cf. A049980, A049981, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A068322, A068323, A127938, A175327, A325328, A325407, A325545, A325546, A325547, A325548.
%K easy,nonn
%O 1,3
%A _Naohiro Nomoto_, Feb 27 2002
%E Extended and edited by _John W. Layman_, Mar 15 2002
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