A104432 -id:A104432 - cp's OEIS frontend

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

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

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

A360492 Square of A(n,m) read by antidiagonals. A(n,m) = number of set partitions of [6n] into 6-element subsets {i, i+k, i+2k, i+3k, i+4k, i+5k} 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, 20, 43, 84, 81, 55, 1, 1, 2, 4, 10, 20, 56, 89, 180, 149, 89, 1, 1, 2, 4, 10, 20, 57, 115, 192, 372, 274, 144, 1

Offset: 1

Peter Dolland, Feb 09 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,   10,   10,   10,   10,    10, ...
  1,   8,  13,   19,   20,   20,   20,   20,    20, ...
  1,  13,  24,   41,   43,   56,   57,   57,    57, ...
  1,  21,  44,   84,   89,  115,  118,  119,   119, ...
  1,  34,  81,  180,  192,  267,  274,  328,   329, ...
  1,  55, 149,  372,  404,  592,  609,  718,   759, ...
  1,  89, 274,  785,  860, 1372, 1416, 1778,  1861, ...
  1, 144, 504, 1637, 1816, 3028, 3136, 3972,  4179, ...
  1, 233, 927, 3442, 3857, 7038, 7323, 9979, 10623, ...
  ...

Columns 1..3 are A000012, A000045(n+1), A000073(n+2).

Cf. A104432, A104443, A360333..A360335, A360491, A360493.

A(n,m) = A104432(n) = A104443(n,6) for m >= floor((6n - 1) / 5).

cp's OEIS Frontend

A104431 Number of ways to split 1, 2, 3, ..., 5n into n arithmetic progressions each with 5 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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A360492 Square of A(n,m) read by antidiagonals. A(n,m) = number of set partitions of [6n] into 6-element subsets {i, i+k, i+2k, i+3k, i+4k, i+5k} with 1 <= k <= m.

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