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%I #52 Sep 30 2019 01:38:11
%S 0,0,1,2,4,7,10,13,19,24,29,37,43,50,62,70,78,92,101,112,129,141,152,
%T 171,185,199,221,237,251,278,293,310,337,356,377,409,427,448,480,505,
%U 525,563,584,609,651,677,700,742,768,800,843,873,899,948,981,1014,1062,1095,1124,1183,1213,1248,1304,1341,1380
%N a(n) is the number of arithmetic progressions of 2 or more positive integers, strictly increasing with sum <= n.
%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.
%F From _Petros Hadjicostas_, Sep 29 2019: (Start)
%F a(n) = Sum_{k = 1..n} A049982(k) = -n + Sum_{k = 1..n} A049980(k) = -n + A049981(k).
%F G.f.: (g.f. of A049982)/(1-x). (End)
%e a(7) = 10 because we have the following arithmetic progressions of two or more positive integers, strictly increasing with sum <= n = 7: 1+2, 1+3, 1+4, 1+5, 1+6, 2+3, 2+4, 2+5, 3+4, and 1+2+3. - _Petros Hadjicostas_, Sep 27 2019
%Y Cf. A014405, A014406, A049980, A049981, A049982, A049986, A049987, A068322, A127938, A175342.
%K nonn
%O 1,4
%A _Clark Kimberling_
%E More terms from _Petros Hadjicostas_, Sep 27 2019
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