Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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