A049981 -id:A049981 - cp's OEIS frontend

A175342 Number of arithmetic progressions (where the difference between adjacent terms is either positive, 0, or negative) of positive integers that sum to n.

1, 2, 4, 5, 6, 10, 8, 10, 15, 14, 12, 22, 14, 18, 28, 21, 18, 34, 20, 28, 38, 28, 24, 46, 31, 32, 48, 38, 30, 62, 32, 40, 58, 42, 46, 73, 38, 46, 68, 58, 42, 84, 44, 56, 90, 56, 48, 94, 55, 70, 90, 66, 54, 106, 70, 74, 100, 70, 60, 130, 62, 74, 118, 81, 82, 130, 68, 84, 120

Offset: 1

Leroy Quet, Apr 17 2010

nonn

			From _Gus Wiseman_, May 15 2019: (Start)
The a(1) = 1 through a(8) = 10 compositions with equal differences:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (12)   (13)    (14)     (15)      (16)       (17)
             (21)   (22)    (23)     (24)      (25)       (26)
             (111)  (31)    (32)     (33)      (34)       (35)
                    (1111)  (41)     (42)      (43)       (44)
                            (11111)  (51)      (52)       (53)
                                     (123)     (61)       (62)
                                     (222)     (1111111)  (71)
                                     (321)                (2222)
                                     (111111)             (11111111)
(End)

Lars Blomberg, Table of n, a(n) for n = 1..10000
Lars Blomberg, C# program for calculating b-file.
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions, Rostok. Math. Kolloq. 64 (2009), 11-16.
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. A000005, A000079, A049980, A049981, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A070211, A127938, A175327, A325328, A325407, A325545, A325546, A325547, A325548, A325557, A325558.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@Differences[#]&]],{n,0,15}] (* returns a(0) = 1, Gus Wiseman, May 15 2019*)

a(n) = 2*A049988(n) - A000005(n).

G.f.: x/(1-x) + Sum_{k>=2} x^k * (1 + x^(k(k-1)/2)) / (1 - x^(k(k-1)/2)) / (1 -x^k).

Edited and extended by Max Alekseyev, May 03 2010

A049980 a(n) is the number of arithmetic progressions of positive integers, strictly increasing with sum n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 4, 7, 6, 6, 9, 7, 8, 13, 9, 9, 15, 10, 12, 18, 13, 12, 20, 15, 15, 23, 17, 15, 28, 16, 18, 28, 20, 22, 33, 19, 22, 33, 26, 21, 39, 22, 26, 43, 27, 24, 43, 27, 33, 44, 31, 27, 50, 34, 34, 49, 34, 30, 60, 31, 36, 57, 38, 40

Offset: 1

Clark Kimberling

nonn

We need to find the number of pairs of positive integers (b, w) so that there is a positive integer m such that m*b + m*(m-1)*w/2 = n. - Petros Hadjicostas, Sep 27 2019

			a(6) = 4 because we have the following strictly increasing arithmetic progressions of positive integers adding up to n = 6: 6, 1+5, 2+4, and 1+2+3. - _Petros Hadjicostas_, Sep 27 2019

Fausto A. C. Cariboni, 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.

Cf. A000217, A014405, A014406, A049981, A049982, A049983, A049986, A049987, A068322, A068323, A068324, A127938, A175342.

Conjecture: a(n) = 1 + Sum_{m|n, m odd > 1} floor(2 * (n - m)/(m* (m - 1))) + Sum_{m|n} floor((n - m * (5 - (-1)^(n/m))/2 + m^2 * (1 - (-1)^(n/m)))/(2*m * (2*m - 1))). - Petros Hadjicostas, Sep 27 2019

G.f.: x/(1-x) + Sum_{k >= 2} x^t(k)/(x^t(k) - x^t(k-1) - x^k + 1) = x/(1-x) + Sum_{k >= 2} 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

A049982 Number of arithmetic progressions of 2 or more positive integers, strictly increasing with sum n.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 3, 3, 6, 5, 5, 8, 6, 7, 12, 8, 8, 14, 9, 11, 17, 12, 11, 19, 14, 14, 22, 16, 14, 27, 15, 17, 27, 19, 21, 32, 18, 21, 32, 25, 20, 38, 21, 25, 42, 26, 23, 42, 26, 32, 43, 30, 26, 49, 33, 33, 48, 33, 29, 59, 30, 35, 56, 37, 39, 61, 33, 39, 58, 49, 35, 67, 36, 42

Offset: 1

Clark Kimberling

nonn

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.
Graeme McRae, Counting arithmetic sequences whose sum is n.
Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
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. A000005, A000217, A014405, A014406, A049980, A049981, A049983, A049986, A049987, A049988, A068322, A127938, A175342.

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

a(n) has generating function x^3/(x^3 - x - x^2 + 1) + x^6/(x^6 - x^3 - x^3 + 1) + x^10/(x^10 - x^6 - x^4 + 1) + ... = Sum_{k >= 2} x^t(k)/(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 positive integers, strictly increasing with sum n. - Graeme McRae, Feb 08 2007
From Petros Hadjicostas, Sep 27 2019: (Start)
a(n) = A049980(n) - 1 = A049988(n) - A000005(n).
a(n) = A049981(n) - A049981(n-1) - 1 for n >= 2.
Conjecture: a(n) = Sum_{m|n, m odd > 1} floor(2 * (n - m)/(m* (m - 1))) + Sum_{m|n} floor((n - m * (5 - (-1)^(n/m))/2 + m^2 * (1 - (-1)^(n/m)))/(2*m * (2*m - 1))).
(End)

More terms from Petros Hadjicostas, Sep 28 2019

A049987 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, 1, 1, 1, 2, 3, 4, 4, 5, 5, 7, 8, 10, 10, 11, 13, 15, 16, 19, 19, 23, 23, 25, 26, 29, 33, 37, 37, 40, 41, 47, 47, 52, 52, 56, 62, 66, 66, 70, 72, 80, 82, 87, 87, 93, 99, 105, 107, 112, 112, 123, 123, 128, 133, 139, 146, 154, 154, 160, 162, 177, 177, 186, 186, 192, 202

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.

Cf. A014405, A014406, A049980, A049981, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A127938.

From Petros Hadjicostas, Sep 29 2019: (Start)
a(n) = Sum_{k = 1..n} A049986(k).
G.f.: (g.f. of A049986)/(1-x). (End)

More terms from Petros Hadjicostas, Sep 29 2019

A049983 a(n) is the number of arithmetic progressions of 2 or more positive integers, strictly increasing with sum <= n.

Original entry on oeis.org

0, 0, 1, 2, 4, 7, 10, 13, 19, 24, 29, 37, 43, 50, 62, 70, 78, 92, 101, 112, 129, 141, 152, 171, 185, 199, 221, 237, 251, 278, 293, 310, 337, 356, 377, 409, 427, 448, 480, 505, 525, 563, 584, 609, 651, 677, 700, 742, 768, 800, 843, 873, 899, 948, 981, 1014, 1062, 1095, 1124, 1183, 1213, 1248, 1304, 1341, 1380

Offset: 1

Clark Kimberling

nonn

			a(7) = 10 because we have the following arithmetic progressions of two or more positive integers, strictly increasing with sum <= n = 7: 1+2, 1+3, 1+4, 1+5, 1+6, 2+3, 2+4, 2+5, 3+4, and 1+2+3. - _Petros Hadjicostas_, Sep 27 2019

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, A049981, A049982, A049986, A049987, A068322, A127938, A175342.

From Petros Hadjicostas, Sep 29 2019: (Start)
a(n) = Sum_{k = 1..n} A049982(k) = -n + Sum_{k = 1..n} A049980(k) = -n + A049981(k).
G.f.: (g.f. of A049982)/(1-x). (End)

More terms from Petros Hadjicostas, Sep 27 2019

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

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

A049991 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, 3, 6, 9, 15, 19, 25, 33, 41, 47, 60, 67, 77, 92, 104, 113, 132, 142, 158, 178, 193, 205, 231, 247, 264, 289, 310, 325, 359, 375, 397, 427, 449, 473, 513, 532, 556, 591, 623, 644, 689, 711, 741, 788, 817, 841, 892, 920, 957, 1003, 1038, 1065, 1121, 1157, 1197, 1248, 1284, 1314, 1384, 1415

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.
Wikipedia, Arithmetic progression.

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

From Petros Hadjicostas, Sep 29 2019: (Start)
a(n) = Sum_{k = 1..n} A049990(k).
G.f.: (g.f. of A049990)/(1-x). (End)

More terms from Petros Hadjicostas, Sep 29 2019

A049992 a(n) is the number of arithmetic progressions of 3 or more positive integers, nondecreasing with sum n.

Original entry on oeis.org

0, 0, 1, 1, 1, 3, 1, 2, 4, 3, 1, 7, 1, 3, 8, 4, 1, 10, 1, 6, 10, 4, 1, 14, 4, 4, 12, 7, 1, 19, 1, 6, 14, 5, 7, 22, 1, 5, 16, 12, 1, 24, 1, 8, 25, 6, 1, 27, 4, 12, 21, 9, 1, 29, 9, 12, 23, 7, 1, 40, 1, 7, 30, 11, 10, 35, 1, 10, 27, 21, 1, 42, 1, 8, 39, 11, 7, 40, 1, 22, 35, 9, 1, 49, 12, 9, 34

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 A049992, 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 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. A007862, A014405, A023645, A049980, A049981, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A049991, A049992, A049994, A175676, A240026, A240027, A307824, A320466.

A049992(n) = (A014405(n)+A023645(n)); \\ (Uses the programs given in A014405 and A023645) - Antti Karttunen, Feb 20 2023

G.f.: Sum_{k>=3} x^k/(1-x^(k*(k-1)/2))/(1-x^k). [Leroy Quet from A049988] - Petros Hadjicostas, Sep 29 2019

a(n) = A014405(n) + A023645(n) = A049994(n) + A175676(n). [Two of the formulas listed by Sequence Machine, both obviously true] - Antti Karttunen, Feb 20 2023

More terms from Petros Hadjicostas, Sep 29 2019

cp's OEIS Frontend

A175342 Number of arithmetic progressions (where the difference between adjacent terms is either positive, 0, or negative) of positive integers that sum to n.

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A049980 a(n) is the number of arithmetic progressions of positive integers, strictly increasing with sum n.

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A049982 Number of arithmetic progressions of 2 or more positive integers, strictly increasing with sum n.

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A049987 a(n) is the number of arithmetic progressions of 4 or more positive integers, strictly increasing with sum <= n.

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A049983 a(n) is the number of arithmetic progressions of 2 or more positive integers, strictly increasing with sum <= n.

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

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

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A049991 a(n) is the number of arithmetic progressions of 2 or more positive integers, nondecreasing with sum <= n.

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