A049988 -id:A049988 - cp's OEIS frontend

A014405 Number of arithmetic progressions of 3 or more positive integers, strictly increasing with sum n.

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, 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, 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

Offset: 1

Clark Kimberling

nonn

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

Antti Karttunen, Table of n, a(n) for n = 1..12580 (first 1000 terms from Fausto A. C. Cariboni)
Sadek Bouroubi and Nesrine 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.
Graeme McRae, Counting arithmetic sequences whose sum is n.
Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
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.
A. N. Pacheco Pulido, Extensiones lineales de un poset y composiciones de números multipartitos, Maestría thesis, Universidad Nacional de Colombia, 2012.
Wikipedia, Arithmetic progression.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

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

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(wRick L. Shepherd, Aug 30 2006

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

a(n) = A049992(n) - A023645(n). - Antti Karttunen, Feb 20 2023

A243815 Number of length n words on alphabet {0,1} such that the length of every maximal block of 0's (runs) is the same.

Original entry on oeis.org

1, 2, 4, 8, 14, 24, 39, 62, 97, 151, 233, 360, 557, 864, 1344, 2099, 3290, 5176, 8169, 12931, 20524, 32654, 52060, 83149, 133012, 213069, 341718, 548614, 881572, 1417722, 2281517, 3673830, 5918958, 9540577, 15384490, 24817031, 40045768, 64637963, 104358789

Offset: 0

Geoffrey Critzer, Jun 11 2014

nonn

Number of terms of A164710 with exactly n+1 binary digits. - Robert Israel, Nov 09 2015
From Gus Wiseman, Jun 23 2025: (Start)
This is the number of subsets of {1..n} with all equal lengths of runs of consecutive elements increasing by 1. For example, the runs of S = {1,2,5,6,8,9} are ((1,2),(5,6),(8,9)), with lengths (2,2,2), so S is counted under a(9). The a(0) = 1 through a(4) = 14 subsets are:
  {}  {}   {}     {}       {}
      {1}  {1}    {1}      {1}
           {2}    {2}      {2}
           {1,2}  {3}      {3}
                  {1,2}    {4}
                  {1,3}    {1,2}
                  {2,3}    {1,3}
                  {1,2,3}  {1,4}
                           {2,3}
                           {2,4}
                           {3,4}
                           {1,2,3}
                           {2,3,4}
                           {1,2,3,4}
(End)

			0110 is a "good" word because the length of both its runs of 0's is 1.
Words of the form 11...1 are good words because the condition is vacuously satisfied.
a(5) = 24 because there are 32 length 5 binary words but we do not count: 00010, 00101, 00110, 01000, 01001, 01100, 10010, 10100.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A164710.
These subsets are ranked by A164707, complement A164708.
For distinct instead of equal lengths we have A384175, complement A384176.
For anti-runs instead of runs we have A384889, for partitions A384888.
For permutations instead of subsets we have A384892, distinct instead of equal A384891.
For partitions instead of subsets we have A384904, strict A384886.
The complement is counted by A385214.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A049988 counts partitions with equal run-lengths, distinct A325325.
A329738 counts compositions with equal run-lengths, distinct A329739.
A384887 counts partitions with equal lengths of gapless runs, distinct A384884.
Cf. A010027, A044813, A069010, A268193, A383013, A384177, A384893.

a:= n-> 1 + add(add((d-> binomial(d+j, d))(n-(i*j-1))
          , j=1..iquo(n+1, i)), i=2..n+1):
seq(a(n), n=0..50);  # Alois P. Heinz, Jun 11 2014

nn=30;Prepend[Map[Total,Transpose[Table[Drop[CoefficientList[Series[ (1+x^k)/(1-x-x^(k+1))-1/(1-x),{x,0,nn}],x],1],{k,1,nn}]]],0]+1
Table[Length[Select[Subsets[Range[n]],SameQ@@Length/@Split[#,#2==#1+1&]&]],{n,0,10}] (* Gus Wiseman, Jun 23 2025 *)

A049986 a(n) is the number of arithmetic progressions of 4 or more positive integers, strictly increasing with sum n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 2, 1, 2, 0, 1, 2, 2, 1, 3, 0, 4, 0, 2, 1, 3, 4, 4, 0, 3, 1, 6, 0, 5, 0, 4, 6, 4, 0, 4, 2, 8, 2, 5, 0, 6, 6, 6, 2, 5, 0, 11, 0, 5, 5, 6, 7, 8, 0, 6, 2, 15, 0, 9, 0, 6, 10, 7, 4, 9, 0, 14, 5, 7, 0, 12, 9, 7, 3

Offset: 1

Clark Kimberling

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12580
Sadek Bouroubi and Nesrine 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.
Jon Maiga, Computer-generated formulas for A049986, Sequence Machine.
Graeme McRae, Counting arithmetic sequences whose sum is n.
Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
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.
A. N. Pacheco Pulido, Extensiones lineales de un poset y composiciones de números multipartitos, Maestría thesis, Universidad Nacional de Colombia, 2012.

Cf. A014405, A014406, A049980, A049981, A049982, A049983, A049987 (partial sums), A049988, A049989, A049990, A049991, A049994, A127938, A321014.

\\ Needs also code from A014405 and A023645:
A049994(n) = (A014405(n) + A023645(n) - if(n%3, 0, n/3));
A321014(n) = (numdiv(n) - 3 + !!(n%2) + !!(n%3)); \\ From A321014.
A049986(n) = (A049994(n)-A321014(n)); \\ Antti Karttunen, Feb 20 2023

G.f.: Sum_{k >= 4} x^t(k)/(x^t(k) - x^t(k-1) - x^k + 1) = Sum_{k >= 4} 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

a(n) = A049994(n) - A321014(n). [Listed by Sequence Machine and obviously true] - Antti Karttunen, Feb 20 2023

A238423 Number of compositions of n avoiding three consecutive parts in arithmetic progression.

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 22, 42, 81, 149, 278, 516, 971, 1812, 3374, 6297, 11770, 21970, 41002, 76523, 142901, 266779, 497957, 929563, 1735418, 3239698, 6047738, 11289791, 21076118, 39344992, 73448769, 137113953, 255965109, 477835991, 892023121, 1665227859

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 26 2014

nonn

These are compositions of n whose second-differences are nonzero. - Gus Wiseman, Jun 03 2019

			The a(5) = 13 such compositions are:
01:  [ 1 1 2 1 ]
02:  [ 1 1 3 ]
03:  [ 1 2 1 1 ]
04:  [ 1 2 2 ]
05:  [ 1 3 1 ]
06:  [ 1 4 ]
07:  [ 2 1 2 ]
08:  [ 2 2 1 ]
09:  [ 2 3 ]
10:  [ 3 1 1 ]
11:  [ 3 2 ]
12:  [ 4 1 ]
13:  [ 5 ]

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..400
Wikipedia, Arithmetic progression

Cf. A238424 (equivalent for partitions).
Cf. A238569 (equivalent for any 3-term arithmetic progression).
Cf. A238686, A295370.
Cf. A007862, A049988, A175342, A325545, A325849, A325851, A325874, A325875.

# b(n, r, d): number of compositions of n where the leftmost part j
#             does not have distance d to the recent part r
b:= proc(n, r, d) option remember; `if`(n=0, 1,
      add(`if`(j=r+d, 0, b(n-j, j, j-r)), j=1..n))
    end:
a:= n-> b(n, infinity, 0):
seq(a(n), n=0..45);

b[n_, r_, d_] := b[n, r, d] = If[n == 0, 1, Sum[If[j == r + d, 0, b[n - j, j, j - r]], {j, 1, n}]]; a[n_] := b[n, Infinity, 0]; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Nov 06 2014, after Maple *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[Differences[#,2],0]&]],{n,0,10}] (* Gus Wiseman, Jun 03 2019 *)

a(n) ~ c * d^n, where d = 1.866800016014240677813344121155900699..., c = 0.540817940878009616510727217687704495... - Vaclav Kotesovec, May 01 2014

A295370 Number of permutations of [n] avoiding three consecutive terms in arithmetic progression.

Original entry on oeis.org

1, 1, 2, 4, 18, 80, 482, 3280, 26244, 231148, 2320130, 25238348, 302834694, 3909539452, 54761642704, 816758411516, 13076340876500, 221396129723368, 3985720881222850, 75503196628737920, 1510373288335622576, 31634502738658957588, 696162960370556156224, 15978760340940405262668

Offset: 0

Alois P. Heinz, Nov 20 2017

nonn

These are permutations of n whose second-differences are nonzero. - Gus Wiseman, Jun 03 2019

			a(3) = 4: 132, 213, 231, 312.
a(4) = 18: 1243, 1324, 1342, 1423, 2134, 2143, 2314, 2413, 2431, 3124, 3142, 3241, 3412, 3421, 4132, 4213, 4231, 4312.

Wikipedia, Arithmetic progression
Index entries related to non-averaging sequences

Column k=0 of A295390.
Cf. A003407, A174080, A238423, A238424, A338765.
Cf. A049988, A175342, A279945, A325545, A325849, A325850, A325851, A325874, A325875.

b:= proc(s, j, k) option remember; `if`(s={}, 1,
      add(`if`(k=0 or 2*j<>i+k, b(s minus {i}, i,
          `if`(2*i-j in s, j, 0)), 0), i=s))
    end:
a:= n-> b({$1..n}, 0$2):
seq(a(n), n=0..12);

Table[Length[Select[Permutations[Range[n]],!MemberQ[Differences[#,2],0]&]],{n,0,5}] (* Gus Wiseman, Jun 03 2019 *)
b[s_, j_, k_] := b[s, j, k] = If[s == {}, 1, Sum[If[k == 0 || 2*j != i + k, b[s~Complement~{i}, i, If[MemberQ[s, 2*i - j ], j, 0]], 0], {i, s}]];
a[n_] := a[n] = b[Range[n], 0, 0];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 16}] (* Jean-François Alcover, Nov 20 2023, after Alois P. Heinz *)

a(22)-a(23) from Vaclav Kotesovec, Mar 22 2022

A307824 Heinz numbers of integer partitions whose augmented differences are all equal.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 11, 13, 15, 16, 17, 19, 23, 29, 31, 32, 37, 41, 43, 47, 53, 55, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 101, 103, 105, 107, 109, 113, 119, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227

Offset: 1

Gus Wiseman, May 03 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
The enumeration of these partitions by sum is given by A129654.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
   11: {5}
   13: {6}
   15: {2,3}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
   37: {12}
   41: {13}
   43: {14}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A049988, A056239, A093641, A112798, A129654, A325327, A325328, A325351, A325359, A325366, A325389, A325394, A325395, A325396.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
aug[y_]:=Table[If[i

A049981 a(n) is the number of arithmetic progressions of positive integers, strictly increasing with sum <= n.

Original entry on oeis.org

1, 2, 4, 6, 9, 13, 17, 21, 28, 34, 40, 49, 56, 64, 77, 86, 95, 110, 120, 132, 150, 163, 175, 195, 210, 225, 248, 265, 280, 308, 324, 342, 370, 390, 412, 445, 464, 486, 519, 545, 566, 605, 627, 653, 696, 723, 747, 790, 817, 850, 894, 925, 952, 1002, 1036, 1070, 1119, 1153, 1183, 1243, 1274, 1310

Offset: 1

Clark Kimberling

nonn

Sadek Bouroubi and Nesrine 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.
Graeme McRae, Counting arithmetic sequences whose sum is n.
Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
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.
A. N. Pacheco Pulido, Extensiones lineales de un poset y composiciones de números multipartitos, Maestría thesis, Universidad Nacional de Colombia, 2012.

Cf. A014405, A014406, A049980, A049982, A049983, A049986, A049987, A068322, A068323, A068324, A127938, A175342.

From Petros Hadjicostas, Sep 29 2019: (Start)
a(n) = Sum_{k = 1..n} A049980(k) = n + Sum_{k = 1..n} A049982(k).
G.f.: (g.f. of A049980)/(1-x). (End)

More terms from Petros Hadjicostas, Sep 29 2019

A014406 Number of strictly increasing arithmetic progressions of positive integers with at least 3 terms and sum <= n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 3, 4, 4, 7, 7, 8, 13, 14, 14, 20, 20, 22, 29, 31, 31, 39, 41, 43, 52, 55, 55, 68, 68, 70, 81, 84, 88, 103, 103, 106, 119, 125, 125, 143, 143, 147, 167, 171, 171, 190, 192, 200, 218, 223, 223, 246, 252, 258, 278, 283, 283, 313, 313, 318, 343, 349, 356, 385, 385

Offset: 1

Clark Kimberling

nonn

			From _Petros Hadjicostas_, Sep 29 2019: (Start)
a(8) = 1 because we have only the following strictly increasing arithmetic progression of positive integers with at least 3 terms and sum <= 8: 1+2+3.
a(9) = 3 because we have the following strictly increasing arithmetic progressions of positive integers with at least 3 terms and sum <= 9: 1+2+3, 1+3+5, and 2+3+4.
a(10) = 4 because we have the following strictly increasing arithmetic progressions of positive integers with at least 3 terms and sum <= 10: 1+2+3, 1+3+5, 2+3+4, and 1+2+3+4.
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..1000
Sadek Bouroubi and Nesrine 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.
Graeme McRae, Counting arithmetic sequences whose sum is n.
Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
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.
A. N. Pacheco Pulido, Extensiones lineales de un poset y composiciones de números multipartitos, Maestría thesis, Universidad Nacional de Colombia, 2012.
Wikipedia, Arithmetic progression.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A007862, A014405, A047966, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A049991, A129654, A240026, A240027, A307824, A320466, A325325, A325328.

a(n) = Sum_{k=1..n} A014405(k). - Sean A. Irvine, Oct 22 2018

G.f.: (g.f. of A014405)/(1-x). - Petros Hadjicostas, Sep 29 2019

a(59)-a(67) corrected by Fausto A. C. Cariboni, Oct 02 2018

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

Original entry on oeis.org

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

Clark Kimberling

nonn

			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].

Clark Kimberling, Table of n, a(n) for n = 1..1000
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.
Graeme McRae, Counting arithmetic sequences whose sum is n.
Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
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.
Wikipedia, Arithmetic progression.

Cf. A014405, A014406, A049980, A049981, A049982, A049983, A049987, A049988, A049989, A049990, A049991, A111333, A127938.

(* 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 *)

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)

A127938 Number of arithmetic progressions of 2 or more nonnegative integers, strictly increasing with sum n.

Original entry on oeis.org

1, 1, 3, 2, 3, 6, 4, 4, 8, 7, 6, 11, 7, 8, 15, 9, 9, 17, 10, 13, 20, 13, 12, 22, 15, 15, 24, 18, 15, 32, 16, 18, 29, 20, 22, 36, 19, 22, 34, 27, 21, 42, 22, 26, 46, 27, 24, 45, 27, 34, 45, 31, 27, 52, 35, 35, 50, 34, 30, 64, 31, 36, 59, 38, 40, 65, 34, 40, 60, 51, 36, 71, 37, 43

Offset: 1

Graeme McRae, Feb 08 2007

nonn

From Petros Hadjicostas, Sep 28 2019: (Start)
We want to find the number of pairs of integers (b, w) such that b >= 0 and w >= 1 and there is an integer m >= 1 so that m*b + (1/2)*m*(m-1)*w = n.
If we insist that b > 0, we get A049982 (= number of arithmetic progressions of 2 or more positive integers, strictly increasing with sum n). The number of integers m >= 1 such that (1/2)*m*(m-1)*w = n equals  A007862(n) (= number of triangular numbers that divide n).
Thus, to get a(n), we add A049982(n) to A007862(n).
(End)

			a(10) = 7 because there are five 2-element arithmetic progressions that sum to 10, as well as 1+2+3+4 and 0+1+2+3+4.

Andrew Howroyd, Table of n, a(n) for n = 1..10000
Sadek Bouroubi and Nesrine 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.
Graeme McRae, Counting arithmetic sequences whose sum is n.
Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
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.
A. N. Pacheco Pulido, Extensiones lineales de un poset y composiciones de números multipartitos, Maestría thesis, Universidad Nacional de Colombia, 2012.
Wikipedia, Arithmetic progression.

Cf. A000217, A007862, A014405, A014406, A049980, A049981, A049982, A049983, A049986, A049987.

seq(n)={Vec(sum(k=1, (sqrtint(8*n+1)-1)\2,  x^binomial(k+1, 2)/(x^binomial(k+2, 2) - x^binomial(k+1, 2) - x^(k+1) + 1) + O(x*x^n)))} \\ Andrew Howroyd, Sep 28 2019

G.f.: x/(x^3 - x - x^2 + 1) + x^3/(x^6 - x^3 - x^3 + 1) + x^6/(x^10 - x^6 - x^4 + 1) + ... = Sum_{k >= 2} x^{t(k-1)}/(x^{t(k)} - x^{t(k-1)} - x^k + 1), where t(k) = A000217(k) is the k-th triangular number. Term k of this generating function generates the number of arithmetic progressions of k nonnegative integers, strictly increasing with sum n.

a(n) = A049982(n) + A007862(n). - Petros Hadjicostas, Sep 28 2019

cp's OEIS Frontend

A014405 Number of arithmetic progressions of 3 or more positive integers, strictly increasing with sum n.

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A243815 Number of length n words on alphabet {0,1} such that the length of every maximal block of 0's (runs) is the same.

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A049986 a(n) is the number of arithmetic progressions of 4 or more positive integers, strictly increasing with sum n.

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A238423 Number of compositions of n avoiding three consecutive parts in arithmetic progression.

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A295370 Number of permutations of [n] avoiding three consecutive terms in arithmetic progression.

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A307824 Heinz numbers of integer partitions whose augmented differences are all equal.

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A049981 a(n) is the number of arithmetic progressions of positive integers, strictly increasing with sum <= n.

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A014406 Number of strictly increasing arithmetic progressions of positive integers with at least 3 terms and sum <= n.

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