A104430 -id:A104430 - 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

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

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).

A332752 The number of permutations of {1,1,1,1,2,2,2,2,...,n,n,n,n} such that each quadruple of k's (k=1..n) is equally spaced with b(k) other elements in between, and b(1) >= b(2) >= ... >= b(n).

Original entry on oeis.org

1, 1, 4, 16, 110, 544, 5444, 32520, 385776, 3282108, 40916528, 354328560, 7200045216, 67347823160, 1182323197504, 18086875471594, 358787259407482, 4564034487662420

Offset: 0

Seiichi Manyama, Feb 22 2020

			In case of n = 1.
     |              | b(1)
-----+--------------+------
   1 | [1, 1, 1, 1] | [0] *
In case of n = 2.
     |                          | b(1),b(2)
-----+--------------------------+----------
   1 | [2, 2, 2, 2, 1, 1, 1, 1] | [0, 0]
   2 | [2, 1, 2, 1, 2, 1, 2, 1] | [1, 1]
   3 | [1, 2, 1, 2, 1, 2, 1, 2] | [1, 1]
   4 | [1, 1, 1, 1, 2, 2, 2, 2] | [0, 0]
In case of n = 3.
     |                                      | b(1),b(2),b(3)
-----+--------------------------------------+---------------
   1 | [3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1] | [0, 0, 0]
   2 | [3, 3, 3, 3, 2, 1, 2, 1, 2, 1, 2, 1] | [1, 1, 0]
   3 | [3, 3, 3, 3, 1, 2, 1, 2, 1, 2, 1, 2] | [1, 1, 0]
   4 | [3, 3, 3, 3, 1, 1, 1, 1, 2, 2, 2, 2] | [0, 0, 0]
   5 | [3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1] | [2, 2, 2]
   6 | [3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2] | [2, 2, 2]
   7 | [2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1] | [2, 2, 2]
   8 | [1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2] | [2, 2, 2]
   9 | [2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3] | [2, 2, 2]
  10 | [1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3] | [2, 2, 2]
  11 | [2, 2, 2, 2, 3, 3, 3, 3, 1, 1, 1, 1] | [0, 0, 0]
  12 | [1, 1, 1, 1, 3, 3, 3, 3, 2, 2, 2, 2] | [0, 0, 0]
  13 | [2, 2, 2, 2, 1, 1, 1, 1, 3, 3, 3, 3] | [0, 0, 0]
  14 | [2, 1, 2, 1, 2, 1, 2, 1, 3, 3, 3, 3] | [1, 1, 0]
  15 | [1, 2, 1, 2, 1, 2, 1, 2, 3, 3, 3, 3] | [1, 1, 0]
  16 | [1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3] | [0, 0, 0]
* (strongly decreasing)

Column 4 of A332762.

Cf. A104430, A261517 (strongly decreasing), A285698, A322178, A332748, A332773, A332783, A332784.

a(10)-a(17) from Max Alekseyev, Sep 27 2023

A337520 Number of set partitions of [4n] into 4-element subsets {i, i+k, i+2k, i+3k} with 1<=k<=n.

Original entry on oeis.org

1, 1, 2, 4, 10, 22, 64, 147, 409, 1092, 3253, 8661, 28585, 83190, 274001, 912373, 3366384, 13253582, 61533277, 290493694

Offset: 0

Alois P. Heinz, Nov 18 2020

			a(4) = 10: {{1,2,3,4}, {5,6,7,8}, {9,10,11,12}, {13,14,15,16}},
  {{1,3,5,7}, {2,4,6,8}, {9,10,11,12}, {13,14,15,16}},
  {{1,2,3,4}, {5,7,9,11}, {6,8,10,12}, {13,14,15,16}},
  {{1,4,7,10}, {2,5,8,11}, {3,6,9,12}, {13,14,15,16}},
  {{1,2,3,4}, {5,6,7,8}, {9,11,13,15}, {10,12,14,16}},
  {{1,3,5,7}, {2,4,6,8}, {9,11,13,15}, {10,12,14,16}},
  {{2,4,6,8}, {1,5,9,13}, {3,7,11,15}, {10,12,14,16}},
  {{1,2,3,4}, {5,8,11,14}, {6,9,12,15}, {7,10,13,16}},
  {{1,3,5,7}, {2,6,10,14}, {9,11,13,15}, {4,8,12,16}},
  {{1,5,9,13}, {2,6,10,14}, {3,7,11,15}, {4,8,12,16}}.

Wikipedia, Partition of a set

Cf. A000566, A014307, A104430, A334250.

Main diagonal of A360333.

b:= proc(s, t) option remember; `if`(s={}, 1, (m-> add(
     `if`({seq(m-h*j, h=1..3)} minus s={}, b(s minus {seq(m-h*j,
      h=0..3)}, t), 0), j=1..min(t, iquo(m-1, 3))))(max(s)))
    end:
a:= proc(n) option remember; forget(b): b({$1..4*n}, n) end:
seq(a(n), n=0..12);

b[s_, t_] := b[s, t] = If[s == {}, 1, Function[m, Sum[       If[Union@Table[m-h*j, {h, 1, 3}] ~Complement~ s == {}, b[s ~Complement~ Union@Table[m-h*j, {h, 0, 3}], t], 0], {j, 1, Min[t, Quotient[m-1, 3]]}]][Max[s]]];
a[n_] := a[n] = b[Range[4n], n];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 12}] (* Jean-François Alcover, Feb 13 2023, after Alois P. Heinz *)

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

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

References

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Crossrefs

Extensions

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.

Views

Author

Keywords

Examples

Crossrefs

Formula

A332752 The number of permutations of {1,1,1,1,2,2,2,2,...,n,n,n,n} such that each quadruple of k's (k=1..n) is equally spaced with b(k) other elements in between, and b(1) >= b(2) >= ... >= b(n).

Views

Author

Keywords

Examples

Crossrefs

Extensions

A337520 Number of set partitions of [4n] into 4-element subsets {i, i+k, i+2k, i+3k} with 1<=k<=n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions