cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

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Author

Jonas Wallgren, Mar 17 2005

Keywords

Examples

			{{{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.

Links

Crossrefs

Cf. A104429, A104431-A104443, A337520.

Programs

  • C
    See Links section.

Extensions

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

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

Views

Author

Peter Dolland, Feb 03 2023

Keywords

Examples

			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, ...
  ...

Crossrefs

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

Formula

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

A334250 Number of set partitions of [3n] into 3-element subsets {i, i+k, i+2k} with 1<=k<=n.

Original entry on oeis.org

1, 1, 2, 4, 12, 35, 129, 567, 2920, 16110, 103467, 717608, 5748214, 47937957, 441139750, 4319093093, 45963368076, 510202534002, 6150655137844, 76789781005325, 1028853084775725, 14294680087131380
Offset: 0

Views

Author

Alois P. Heinz, Apr 20 2020

Keywords

Comments

Differs from A331621 first at n=7.

Examples

			a(2) = 2: 123|456, 135|246.
a(3) = 4: 123|456|789, 123|468|579, 135|246|789, 147|258|369.

Links

Crossrefs

Cf. A014307 (the same for 2-element subsets), A025035, A059108, A104429 (where k is not restricted), A285527, A331621, A337520.
Main diagonal of A360334.

Programs

  • Maple
    b:= proc(s, t) option remember; `if`(s={}, 1, (m-> add(
     `if`({m-j, m-2*j} minus s={}, b(s minus {m, m-j, m-2*j},
            t), 0), j=1..min(t, iquo(m-1, 2))))(max(s)))
    end:
a:= proc(n) option remember; forget(b): b({$1..3*n}, n) end:
seq(a(n), n=0..12);
  • Mathematica
    b[s_List, t_] := b[s, t] = If[s == {}, 1, Function[m, Sum[If[{m - j, m - 2j} ~Complement~ s == {}, b[s ~Complement~ {m, m - j, m - 2j}, t], 0], {j, 1, Min[t, Quotient[m - 1, 2]]}]][Max[s]]];
a[n_] := a[n] = b[Range[3n], n];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 12}] (* Jean-François Alcover, May 10 2020, after Maple *)

Formula

a(n) <= A104429(n) <= A025035(n).

Extensions

a(17)-a(21) from Martin Fuller, Jul 19 2025

A349430 Number of set partitions of [5n] into 5-element subsets {i, i+k, i+2k, i+3k, i+4k} with 1<=k<=n.

Original entry on oeis.org

1, 1, 2, 4, 10, 20, 58, 124, 344, 811, 2071, 4973, 15454, 36031, 96212, 237563, 668695, 1626751, 4674373, 11470722, 31460456, 81705943, 224598113
Offset: 0

Views

Author

Alois P. Heinz, Nov 17 2021

Keywords

Examples

			a(4) = 10: {{1,2,3,4,5}, {6,7,8,9,10}, {11,12,13,14,15}, {16,17,18,19,20}},
  {{1,3,5,7,9}, {2,4,6,8,10}, {11,12,13,14,15}, {16,17,18,19,20}},
  {{1,2,3,4,5}, {6,8,10,12,14}, {7,9,11,13,15}, {16,17,18,19,20}},
  {{1,4,7,10,13}, {2,5,8,11,14}, {3,6,9,12,15}, {16,17,18,19,20}},
  {{1,2,3,4,5}, {6,7,8,9,10}, {11,13,15,17,19}, {12,14,16,18,20}},
  {{1,3,5,7,9}, {2,4,6,8,10}, {11,13,15,17,19}, {12,14,16,18,20}},
  {{1,5,9,13,17}, {2,4,6,8,10}, {3,7,11,15,19}, {12,14,16,18,20}},
  {{1,2,3,4,5}, {6,9,12,15,18}, {7,10,13,16,19}, {8,11,14,17,20}},
  {{1,3,5,7,9}, {2,6,10,14,18}, {4,8,12,16,20}, {11,13,15,17,19}},
  {{1,5,9,13,17}, {2,6,10,14,18}, {3,7,11,15,19}, {4,8,12,16,20}}.

Links

Crossrefs

Cf. A000567 (number of subsets), A008587 (number of elements), A104431 (when k is unbounded), A337520.
Main diagonal of A360491.

Programs

  • Maple
    b:= proc(s, t) option remember; `if`(s={}, 1, (m-> add(
     `if`({seq(m-h*j, h=1..4)} minus s={}, b(s minus {seq(m-h*j,
      h=0..4)}, t), 0), j=1..min(t, iquo(m-1, 4))))(max(s)))
    end:
a:= proc(n) option remember; forget(b): b({$1..5*n}, n) end:
seq(a(n), n=0..10);
  • Mathematica
    b[s_, t_] := b[s, t] = If[s == {}, 1, Function[m, Sum[If[Union[Table[m - h*j, {h, 1, 4}] ~Complement~ s] == {}, b[s  ~Complement~ Union[Table[m - h*j, {h, 0, 4}]], t], 0], {j, 1, Min[t, Quotient[m-1, 4]]}]][Max[s]]];
a[n_] := a[n] = b[Range[5n], n];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 15}] (* Jean-François Alcover, May 16 2022, after Alois P. Heinz *)

Extensions

a(22) from Alois P. Heinz, Nov 23 2022
Showing 1-4 of 4 results.

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