A334250 -id:A334250 - 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

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 *)

A360334 Array read by antidiagonals downwards: A(n,m) = number of set partitions of [3n] into 3-element subsets {i, i+k, i+2k} 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, 5, 7, 8, 1, 1, 2, 5, 12, 13, 13, 1, 1, 2, 5, 15, 25, 24, 21, 1, 1, 2, 5, 15, 35, 56, 44, 34, 1, 1, 2, 5, 15, 46, 84, 126, 81, 55, 1, 1, 2, 5, 15, 55, 129, 211, 281, 149, 89, 1, 1, 2, 5, 15, 55, 185, 346, 537, 625, 274, 144, 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,   5,    5,    5,    5,     5,     5, ...
  1,  5,   7,  12,   15,   15,   15,    15,    15, ...
  1,  8,  13,  25,   35,   46,   55,    55,    55, ...
  1, 13,  24,  56,   84,  129,  185,   232,   232, ...
  1, 21,  44, 126,  211,  346,  567,   831,  1040, ...
  1, 34,  81, 281,  537,  973, 1781,  2920,  4242, ...
  1, 55, 149, 625, 1352, 2732, 5643, 10213, 16110, ...
  ...

Main diagonal is A334250.
Columns 1..3 are A000012, A000045(n+1), A000073(n+2).
Cf. A104429, A104443, A360333, A360335.

A(n,m) = A104429(n) = A104443(n,3) for m >= floor((3n - 1) / 2).

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

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A360334 Array read by antidiagonals downwards: A(n,m) = number of set partitions of [3n] into 3-element subsets {i, i+k, i+2k} with 1 <= k <= m.

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