login
Number of arithmetic progressions of 3 or more positive integers, strictly increasing with sum n.
18

%I #51 May 01 2024 17:02:34

%S 0,0,0,0,0,1,0,0,2,1,0,3,0,1,5,1,0,6,0,2,7,2,0,8,2,2,9,3,0,13,0,2,11,

%T 3,4,15,0,3,13,6,0,18,0,4,20,4,0,19,2,8,18,5,0,23,6,6,20,5,0,30,0,5,

%U 25,6,7,29,0,6,24,15,0,32,0,6,34,7,4,34,0,14,31,7,0,39,9,7,31,9,0,49,5,9,33,8,10,42,0,12

%N Number of arithmetic progressions of 3 or more positive integers, strictly increasing with sum n.

%H Antti Karttunen, <a href="/A014405/b014405.txt">Table of n, a(n) for n = 1..12580</a> (first 1000 terms from Fausto A. C. Cariboni)

%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="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 G.f.: Sum_{k >= 3} x^t(k)/(x^t(k) - x^t(k-1) - x^k + 1) = Sum_{k >= 3} x^t(k)/((1 - x^k) * (1 - x^t(k-1))), where t(k) = k*(k+1)/2 = A000217(k) is the k-th triangular number [_Graeme McRae_]. - _Petros Hadjicostas_, Sep 29 2019

%F a(n) = A049992(n) - A023645(n). - _Antti Karttunen_, Feb 20 2023

%e E.g., 15 = 1+2+3+4+5 = 1+5+9 = 2+5+8 = 3+5+7 = 4+5+6.

%o (PARI) a(n)= t=0; st=0; forstep(s=(n-3)\3,1,-1, st++; for(c=1,st, m=3; w=m*(s+c); while(w<n, w=w+s+m*c; m++); if(w==n, t++))); t \\ _Rick L. Shepherd_, Aug 30 2006

%Y Cf. A000217, A007862, A014406, A014407, A023645, A047966, A049982, A049983, A049986, A049987, A049992, A129654, A240026, A240027, A307824, A320466, A325325, A325328.

%K nonn

%O 1,9

%A _Clark Kimberling_