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A049994 a(n) is the number of arithmetic progressions of 4 or more positive integers, nondecreasing with sum n.

%I A049994 #48 Feb 20 2023 07:51:25
%S A049994 0,0,0,1,1,1,1,2,1,3,1,3,1,3,3,4,1,4,1,6,3,4,1,6,4,4,3,7,1,9,1,6,3,5,
%T A049994 7,10,1,5,3,12,1,10,1,8,10,6,1,11,4,12,4,9,1,11,9,12,4,7,1,20,1,7,9,
%U A049994 11,10,13,1,10,4,21,1,18,1,8,14,11,7,14,1,22,8,9,1,21,12,9,5,15,1,29,8
%N A049994 a(n) is the number of arithmetic progressions of 4 or more positive integers, nondecreasing with sum n.
%H A049994 Antti Karttunen, <a href="/A049994/b049994.txt">Table of n, a(n) for n = 1..12580</a>
%H A049994 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 A049994 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 A049994 Jon Maiga, <a href="http://sequencedb.net/s/A049994">Computer-generated formulas for A049994</a>, Sequence Machine.
%H A049994 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 A049994 Graeme McRae, <a href="/A049988/a049988.pdf">Counting arithmetic sequences whose sum is n</a> [Cached copy]
%H A049994 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 A049994 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 A049994 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 A049994 Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic_progression">Arithmetic progression</a>.
%F A049994 G.f.: Sum_{k >= 4} x^k/(1-x^(k*(k-1)/2))/(1-x^k). [_Leroy Quet_ from A049988] - _Petros Hadjicostas_, Sep 29 2019
%F A049994 a(n) = A049992(n) - A175676(n) = A049986(n) + A321014(n). [Two of the formulas listed by Sequence Machine, both obviously true] - _Antti Karttunen_, Feb 20 2023
%o A049994 (PARI) A049994(n) = (A049992(n)-if(n%3, 0, n/3)); \\ _Antti Karttunen_, Feb 20 2023
%Y A049994 Cf.  A014405, A014406, A175676, A049980, A049981, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A049991, A049992, A049993, A127938, A321014.
%K A049994 nonn
%O A049994 1,8
%A A049994 _Clark Kimberling_
%E A049994 More terms from _Petros Hadjicostas_, Sep 29 2019