A104443 -id:A104443 - cp's OEIS frontend

A104429 Number of ways to split {1, 2, 3, ..., 3n} into n arithmetic progressions each with 3 terms.

1, 1, 2, 5, 15, 55, 232, 1161, 6643, 44566, 327064, 2709050, 24312028, 240833770, 2546215687, 29251369570, 355838858402, 4658866773664, 64127566159756, 940320691236206

Offset: 0

Jonas Wallgren, Mar 17 2005

			{{{1,2,3},{4,5,6},{7,8,9}}, {{1,2,3},{4,6,8},{5,7,9}}, {{1,3,5},{2,4,6},{7,8,9}}, {{1,4,7},{2,5,8},{3,6,9}}, {{1,5,9},{2,3,4},{6,7,8}}} are the 5 ways to split 1, 2, 3, ..., 9 into 3 arithmetic progressions each with 3 elements. Thus a(3)=5.

R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, Univ. Calgary, Dept. Mathematics, Research Paper No. 129, 1971.
R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, in Proc. Conf. Number Theory. Pullman, WA, 1971, pp. 221-223.
R. K. Guy, Packing [1,n] with solutions of ax + by = cz; the unity of combinatorics, in Colloq. Internaz. Teorie Combinatorie. Rome, 1973, Atti Conv. Lincei. Vol. 17, Part II, pp. 173-179, 1976.

R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission]. See sequence "M".
Christian Hercher and Frank Niedermeyer, Efficient Calculation the Number of Partitions of the Set {1, 2, ..., 3n} into Subsets {x, y, z} Satisfying x + y = z, arXiv:2307.00303 [math.CO], 2023.
R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.] Gives a(0)-a(10).

Cf. A104430-A104443.
All of A279197, A279198, A202705, A279199, A282615 are concerned with counting solutions to X+Y=2Z in various ways.
See also A002848, A002849, A334250.

a(11)-a(14) from Alois P. Heinz, Dec 28 2011
a(15)-a(17) from Fausto A. C. Cariboni, Feb 22 2017
a(18)-a(19) from Martin Fuller, Jul 08 2025

A104430 Number of ways to split 1, 2, 3, ..., 4n into n arithmetic progressions each with 4 terms.

Original entry on oeis.org

1, 1, 2, 4, 11, 23, 68, 161, 488, 1249, 3771, 10388, 35725, 110449, 387057, 1411784, 5938390, 26054261, 129231034, 708657991

Offset: 0

Jonas Wallgren, Mar 17 2005

			{{{1,2,3,4},{5,6,7,8},{9,10,11,12}}, {{1,2,3,4},{5,7,9,11},{6,8,10,12}}, {{1,3,5,7},{2,4,6,8},{9,10,11,12}}, {{1,4,7,10},{2,5,8,11},{3,6,9,12}}} are the 4 ways to split 1, 2, 3, ..., 12 into 3 arithmetic progressions each with 4 terms. Thus a(3)=4.

Rémy Sigrist, C program

Cf. A104429, A104431-A104443, A337520.

C
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a(11)-a(17) from Alois P. Heinz, Dec 28 2011
a(0)=1 prepended by Alois P. Heinz, Nov 18 2020
a(18)-a(19) from Rémy Sigrist, Feb 07 2022

A104431 Number of ways to split 1, 2, 3, ..., 5n into n arithmetic progressions each with 5 terms.

Original entry on oeis.org

1, 1, 2, 4, 10, 21, 59, 125, 349, 848, 2224, 5210, 15720, 37096, 98241, 245251, 684475, 1703174, 4915084, 12024901, 33594399

Offset: 0

Jonas Wallgren, Mar 17 2005

Cf. A104429-A104430, A104432-A104443, A349430.

a(11)-a(18) from Alois P. Heinz, Dec 28 2011

a(19)-a(20) from Alois P. Heinz, Nov 18 2021

A104442 Number of ways to split 1, 2, 3, ..., tn into n arithmetic progressions each with t terms, t>n.

Original entry on oeis.org

1, 1, 2, 4, 10, 20, 56, 116, 320, 736, 1872, 4176, 12712, 28368, 73592, 177272, 487304, 1130712, 3209792, 7494720, 20419744, 49706280, 129803592, 311179624

Offset: 0

Jonas Wallgren, Mar 17 2005

Cf. A104429-A104441, A104443.

a(0), a(13)-a(23) from Alois P. Heinz, Nov 18 2020

A360333 Array read by antidiagonals downwards: A(n,m) = number of set partitions of [4n] into 4-element subsets {i, i+k, i+2k, i+3k} with 1 <= k <= m.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 5, 1, 1, 2, 4, 7, 8, 1, 1, 2, 4, 10, 13, 13, 1, 1, 2, 4, 11, 19, 24, 21, 1, 1, 2, 4, 11, 22, 41, 44, 34, 1, 1, 2, 4, 11, 23, 48, 84, 81, 55, 1, 1, 2, 4, 11, 23, 64, 101, 180, 149, 89, 1

Offset: 1

Peter Dolland, Feb 03 2023

			Square array begins:
  1,  1,   1,   1,   1,   1,   1,   1,    1, ...
  1,  2,   2,   2,   2,   2,   2,   2,    2, ...
  1,  3,   4,   4,   4,   4,   4,   4,    4, ...
  1,  5,   7,  10,  11,  11,  11,  11,   11, ...
  1,  8,  13,  19,  22,  23,  23,  23,   23, ...
  1, 13,  24,  41,  48,  64,  68,  68,   68, ...
  1, 21,  44,  84, 101, 134, 147, 148,  161, ...
  1, 34,  81, 180, 225, 318, 353, 409,  444, ...
  1, 55, 149, 372, 485, 721, 814, 929, 1092, ...
  ...

Main diagonal is A337520.
Columns 1..3 are A000012, A000045(n+1), A000073(n+2).
Cf. A104430, A104443, A360334, A360335.

A(n,m) = A104430(n) = A104443(n,4) for m >= floor((4n - 1) / 3).

A360335 Array read by antidiagonals downwards: A(n,m) = number of set partitions of [2n] into 2-element subsets {i, i+k} with 1 <= k <= m.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 3, 7, 5, 1, 1, 3, 12, 16, 8, 1, 1, 3, 15, 35, 38, 13, 1, 1, 3, 15, 63, 105, 89, 21, 1, 1, 3, 15, 90, 226, 329, 209, 34, 1, 1, 3, 15, 105, 417, 841, 1014, 491, 55, 1, 1, 3, 15, 105, 645, 1787, 3251, 3116, 1153, 89, 1

Offset: 1

Peter Dolland, Feb 03 2023

			Square array begins:
  1,  1,    1,    1,     1,      1,      1,       1,       1, ...
  1,  2,    3,    3,     3,      3,      3,       3,       3, ...
  1,  3,    7,   12,    15,     15,     15,      15,      15, ...
  1,  5,   16,   35,    63,     90,    105,     105,     105, ...
  1,  8,   38,  105,   226,    417,    645,     840,     945, ...
  1, 13,   89,  329,   841,   1787,   3348,    5445,    7665, ...
  1, 21,  209, 1014,  3251,   7938,  16717,   31647,   53250, ...
  1, 34,  491, 3116, 12483,  36500,  86311,  180560,  344403, ...
  1, 55, 1153, 9610, 47481, 167631, 459803, 1062435, 2211181, ...
  ...

Main diagonal is A014307.
Columns 1..4 are A000012, A000045(n+1), A052967, A320346.
Cf. A001147, A104443, A360333, A360334.

A(n,m) = A001147(n) = A104443(n,2) for m >= 2n - 1.

A104432 Number of ways to split 1, 2, 3, ..., 6n into n arithmetic progressions each with 6 terms.

Original entry on oeis.org

1, 1, 2, 4, 10, 20, 57, 119, 329, 760, 1942, 4452, 13574, 30665, 80117, 194856, 540694

Offset: 0

Jonas Wallgren, Mar 17 2005

nonn

Cf. A104429-A104431, A104433-A104443.

a(0), a(11)-a(16) from Alois P. Heinz, Nov 18 2020

A104433 Number of ways to split 1, 2, 3, ..., 7n into n arithmetic progressions each with 7 terms.

Original entry on oeis.org

1, 1, 2, 4, 10, 20, 56, 117, 323, 745, 1896, 4242, 12883, 29108, 75725, 183366, 504215, 1176776

Offset: 0

Jonas Wallgren, Mar 17 2005

Cf. A104429-A104432, A104434-A104443.

a(0), a(11)-a(17) from Alois P. Heinz, Nov 18 2020

A104435 Number of ways to split 1, 2, 3, ..., 2n into 2 arithmetic progressions each with n terms.

Original entry on oeis.org

1, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Jonas Wallgren, Mar 17 2005

The common difference in an arithmetic progression must be a positive integer. - David A. Corneth, Apr 14 2024

			From _R. J. Mathar_, Apr 14 2024: (Start)
a(2)=3 offers 3 ways of splitting (1,2,3,4): {(1,2),(3,4)}, {(1,3),(2,4)}, {(1,4),(2,3)}.
a(n)=2 for n>=3 because there are at least the two ways of splitting (1,2,..,2n) into the even and odd numbers. (End)

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A104429, A104430, A104431, A104432, A104433, A104434, A104436, A104437, A104438, A104439, A104440, A104441, A104442, A104443.

a(n) = if(n <= 2, [1,3][n], 2) \\ David A. Corneth, Apr 14 2024

a(1) = 1, a(2) = 3, a(n) = 2 for n >= 3. Proof of the latter: if the common difference in an arithmetic progression, starting with a number at least 1, is at least 3 then the largest term in that arithmetic progression is at least 1 + 3*(n-1) = 3*n - 2. But 3*n - 2 > 2*n for n > 2. - David A. Corneth, Apr 14 2024

G.f.: x*(1 + 2*x - x^2)/(1 - x). - Stefano Spezia, Apr 14 2024

More terms from David A. Corneth, Apr 14 2024

A104437 Number of ways to split 1, 2, 3, ..., 4n into 4 arithmetic progressions each with n terms.

Original entry on oeis.org

1, 105, 15, 11, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 1

Jonas Wallgren, Mar 17 2005

nonn

Cf. A104429-A104436, A104438-A104443.

cp's OEIS Frontend

A104429 Number of ways to split {1, 2, 3, ..., 3n} into n arithmetic progressions each with 3 terms.

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A104430 Number of ways to split 1, 2, 3, ..., 4n into n arithmetic progressions each with 4 terms.

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C

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A104431 Number of ways to split 1, 2, 3, ..., 5n into n arithmetic progressions each with 5 terms.

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A104442 Number of ways to split 1, 2, 3, ..., tn into n arithmetic progressions each with t terms, t>n.

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A360333 Array read by antidiagonals downwards: A(n,m) = number of set partitions of [4n] into 4-element subsets {i, i+k, i+2k, i+3k} with 1 <= k <= m.

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A360335 Array read by antidiagonals downwards: A(n,m) = number of set partitions of [2n] into 2-element subsets {i, i+k} with 1 <= k <= m.

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A104432 Number of ways to split 1, 2, 3, ..., 6n into n arithmetic progressions each with 6 terms.

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A104433 Number of ways to split 1, 2, 3, ..., 7n into n arithmetic progressions each with 7 terms.

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A104435 Number of ways to split 1, 2, 3, ..., 2n into 2 arithmetic progressions each with n terms.

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PARI

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A104437 Number of ways to split 1, 2, 3, ..., 4n into 4 arithmetic progressions each with n terms.

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