cp's OEIS Frontend

A049990 a(n) is the number of arithmetic progressions of 2 or more positive integers, nondecreasing with sum n.

%I A049990 #43 Sep 30 2019 02:02:22
%S A049990 0,1,2,3,3,6,4,6,8,8,6,13,7,10,15,12,9,19,10,16,20,15,12,26,16,17,25,
%T A049990 21,15,34,16,22,30,22,24,40,19,24,35,32,21,45,22,30,47,29,24,51,28,37,
%U A049990 46,35,27,56,36,40,51,36,30,70,31,38,61,43
%N A049990 a(n) is the number of arithmetic progressions of 2 or more positive integers, nondecreasing with sum n.
%H A049990 Clark Kimberling, <a href="/A049990/b049990.txt">Table of n, a(n) for n = 1..1000</a>
%H A049990 Sadek Bourbaki and Nevrine 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 A049990 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 A049990 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 A049990 Graeme McRae, <a href="/A049988/a049988.pdf">Counting arithmetic sequences whose sum is n</a> [Cached copy]
%H A049990 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 A049990 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 A049990 Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic_progression">Arithmetic progression</a>.
%F A049990 a(A000040(n)) = A111333(n). - _Clark Kimberling_, Dec 26 2016
%F A049990 From _Petros Hadjicostas_, Sep 29 2019: (Start)
%F A049990 a(n) = A049988(n) - 1. [Note that A049988 has offset 0.]
%F A049990 G.f.: Sum_{k>=2} x^k/(1-x^(k*(k-1)/2))/(1-x^k). [_Leroy Quet_ from A049988]
%F A049990 (End)
%e A049990 a(6) counts these 6 partitions of 6: [5,1], [4,2], [3,3], [3,2,1], [2,2,2], [1,1,1,1,1,1].
%t A049990 (* Program 1 *)
%t A049990 Map[Length[Map[#[[2]] &, Select[Map[{Apply[SameQ, Differences[#]], #} &,
%t A049990 IntegerPartitions[#]], #[[1]] &]]] &, Range[40]] - 1
%t A049990 (* _Peter J. C. Moses_, Dec 24 2016 *)
%t A049990 (* Program 2 *)
%t A049990 enumerateArithmeticPartitions[n_] := Module[{allDivs, oddDivs},
%t A049990 {allDivs, oddDivs} = {#, Select[#, OddQ]} &[Divisors[n]]; Map[Reverse, Union[Flatten[Table[If[OddQ[cDiff], (Flatten[
%t A049990 Map[{If[(2 n - #) cDiff <= # (# - 2), {Table[(cDiff + # - 2 cDiff n/#)/2 +
%t A049990 cDiff term, {term, 0, 2 n/# - 1}]}, {}], If[# (# - 1) cDiff <= 2 (n - #),
%t A049990 {Table[(cDiff + 2 n/# - # cDiff)/2 + cDiff term, {term, 0, # - 1}]},
%t A049990 {}]} &, oddDivs], 2]), (Flatten[Map[If[(n - #) cDiff <= 2 # (# - 1),
%t A049990 {Table[(cDiff + 2 # - n cDiff/#)/2 + cDiff term, {term, 0, n/# - 1}]}, {}] &,
%t A049990 allDivs], 1])], {cDiff, 0, n - 2}], 1]]]];
%t A049990 Join[{0}, Map[Length[enumerateArithmeticPartitions[#]] - 1 &, Range[2, 300]]]
%t A049990 n = 12; enumerateArithmeticPartitions[12] (* shows the desired partition of n *)
%t A049990 (* _Peter J. C. Moses_, Dec 24 2016 *)
%Y A049990 Cf. A014405, A014406, A049980, A049981, A049982, A049983, A049987, A049988, A049989, A049990, A049991, A111333, A127938.
%K A049990 nonn
%O A049990 1,3
%A A049990 _Clark Kimberling_