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 1-6 of 6 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

Views

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

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

Views

Author

Jonas Wallgren, Mar 17 2005

Keywords

Crossrefs

Cf. A104429-A104431, A104433-A104443.

Extensions

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

A332773 The number of permutations of {1,1,1,1,1,2,2,2,2,2,...,n,n,n,n,n} with the property that b(1) >= b(2) >= ... >= b(n), where five k's are skipped by b(k) for k=1..n.

Original entry on oeis.org

1, 1, 4, 16, 104, 508, 5136, 28224, 333360, 2793888, 34010208, 276522240, 5903380896, 50068045536, 892740867024, 13555604385504, 260760485969664, 3084227796562768, 91112167715233008, 1145087508241888160
Offset: 0

Views

Author

Seiichi Manyama, Feb 24 2020

Keywords

Examples

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

Crossrefs

Column 5 of A332762.
Cf. A104431, A322178, A332748, A332752, A332783, A332784.

Extensions

a(10)-a(19) from Max Alekseyev, Sep 26 2023

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

Views

Author

Jonas Wallgren, Mar 17 2005

Keywords

Comments

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

Examples

			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)

Links

Crossrefs

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

Programs

Formula

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

Extensions

More terms from David A. Corneth, Apr 14 2024

A360491 Square of A(n,m) read by antidiagonals. A(n,m) = number of set partitions of [5n] into 5-element subsets {i, i+k, i+2k, i+3k, i+4k} 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, 10, 19, 24, 21, 1, 1, 2, 4, 10, 20, 41, 44, 34, 1, 1, 2, 4, 10, 21, 43, 84, 81, 55, 1, 1, 2, 4, 10, 21, 58, 89, 180, 149, 89, 1, 1, 2, 4, 10, 21, 59, 120, 192, 372, 274, 144, 1
Offset: 1

Views

Author

Peter Dolland, Feb 09 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,   10,   10,   10,   10,   10, ...
  1,   8,  13,   19,   20,   21,   21,   21,   21, ...
  1,  13,  24,   41,   43,   58,   59,   59,   59, ...
  1,  21,  44,   84,   89,  120,  124,  125,  125, ...
  1,  34,  81,  180,  192,  280,  289,  344,  349, ...
  1,  55, 149,  372,  404,  626,  648,  759,  811, ...
  1,  89, 274,  785,  860, 1454, 1510, 1877, 1996, ...
  1, 144, 504, 1637, 1816, 3272, 3414, 4263, 4565, ...
  ...

Crossrefs

Main diagonal is A349430.
Columns 1..3 are A000012, A000045(n+1), A000073(n+2).
Cf. A104431, A104443, A360333..A360335, A360492, A360493.

Formula

A(n,m) = A104431(n) = A104443(n,5) for m >= floor((5n - 1) / 4).

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 1-6 of 6 results.

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