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%I #38 Sep 30 2019 01:38:26
%S 0,0,0,0,0,1,1,1,3,4,4,7,7,8,13,14,14,20,20,22,29,31,31,39,41,43,52,
%T 55,55,68,68,70,81,84,88,103,103,106,119,125,125,143,143,147,167,171,
%U 171,190,192,200,218,223,223,246,252,258,278,283,283,313,313,318,343,349,356,385,385
%N Number of strictly increasing arithmetic progressions of positive integers with at least 3 terms and sum <= n.
%H Fausto A. C. Cariboni, <a href="/A014406/b014406.txt">Table of n, a(n) for n = 1..1000</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="http://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="http://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 a(n) = Sum_{k=1..n} A014405(k). - _Sean A. Irvine_, Oct 22 2018
%F G.f.: (g.f. of A014405)/(1-x). - _Petros Hadjicostas_, Sep 29 2019
%e From _Petros Hadjicostas_, Sep 29 2019: (Start)
%e a(8) = 1 because we have only the following strictly increasing arithmetic progression of positive integers with at least 3 terms and sum <= 8: 1+2+3.
%e a(9) = 3 because we have the following strictly increasing arithmetic progressions of positive integers with at least 3 terms and sum <= 9: 1+2+3, 1+3+5, and 2+3+4.
%e a(10) = 4 because we have the following strictly increasing arithmetic progressions of positive integers with at least 3 terms and sum <= 10: 1+2+3, 1+3+5, 2+3+4, and 1+2+3+4.
%e (End)
%Y Cf. A007862, A014405, A047966, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A049991, A129654, A240026, A240027, A307824, A320466, A325325, A325328.
%K nonn
%O 1,9
%A _Clark Kimberling_
%E a(59)-a(67) corrected by _Fausto A. C. Cariboni_, Oct 02 2018
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