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A049990
a(n) is the number of arithmetic progressions of 2 or more positive integers, nondecreasing with sum n.
14
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, 21, 15, 34, 16, 22, 30, 22, 24, 40, 19, 24, 35, 32, 21, 45, 22, 30, 47, 29, 24, 51, 28, 37, 46, 35, 27, 56, 36, 40, 51, 36, 30, 70, 31, 38, 61, 43
OFFSET
1,3
LINKS
Sadek Bourbaki and Nevrine Benyahia Tani, Integer partitions into arithmetic progressions, Rostok. Math. Kolloq. 64 (2009), 11-16.
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions with an odd common difference, Integers 9(1) (2009), 77-81.
Augustine O. Munagi, Combinatorics of integer partitions in arithmetic progression, Integers 10(1) (2010), 73-82.
Augustine O. Munagi and Temba Shonhiwa, On the partitions of a number into arithmetic progressions, Journal of Integer Sequences 11 (2008), Article 08.5.4.
FORMULA
a(A000040(n)) = A111333(n). - Clark Kimberling, Dec 26 2016
From Petros Hadjicostas, Sep 29 2019: (Start)
a(n) = A049988(n) - 1. [Note that A049988 has offset 0.]
G.f.: Sum_{k>=2} x^k/(1-x^(k*(k-1)/2))/(1-x^k). [Leroy Quet from A049988]
(End)
EXAMPLE
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].
MATHEMATICA
(* Program 1 *)
Map[Length[Map[#[[2]] &, Select[Map[{Apply[SameQ, Differences[#]], #} &,
IntegerPartitions[#]], #[[1]] &]]] &, Range[40]] - 1
(* Peter J. C. Moses, Dec 24 2016 *)
(* Program 2 *)
enumerateArithmeticPartitions[n_] := Module[{allDivs, oddDivs},
{allDivs, oddDivs} = {#, Select[#, OddQ]} &[Divisors[n]]; Map[Reverse, Union[Flatten[Table[If[OddQ[cDiff], (Flatten[
Map[{If[(2 n - #) cDiff <= # (# - 2), {Table[(cDiff + # - 2 cDiff n/#)/2 +
cDiff term, {term, 0, 2 n/# - 1}]}, {}], If[# (# - 1) cDiff <= 2 (n - #),
{Table[(cDiff + 2 n/# - # cDiff)/2 + cDiff term, {term, 0, # - 1}]},
{}]} &, oddDivs], 2]), (Flatten[Map[If[(n - #) cDiff <= 2 # (# - 1),
{Table[(cDiff + 2 # - n cDiff/#)/2 + cDiff term, {term, 0, n/# - 1}]}, {}] &,
allDivs], 1])], {cDiff, 0, n - 2}], 1]]]];
Join[{0}, Map[Length[enumerateArithmeticPartitions[#]] - 1 &, Range[2, 300]]]
n = 12; enumerateArithmeticPartitions[12] (* shows the desired partition of n *)
(* Peter J. C. Moses, Dec 24 2016 *)
KEYWORD
nonn
STATUS
approved