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.

A104443 Square of P(n,t) read by antidiagonals. P(n,t) = number of ways to split [t*n] into n arithmetic progressions each with t terms.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 2, 15, 1, 1, 2, 5, 105, 1, 1, 2, 4, 15, 945, 1, 1, 2, 4, 11, 55, 10395, 1, 1, 2, 4, 10, 23, 232, 135135, 1, 1, 2, 4, 10, 21, 68, 1161, 2027025, 1, 1, 2, 4, 10, 20, 59, 161, 6643, 34459425, 1, 1, 2, 4, 10, 20, 57, 125, 488, 44566, 654729075, 1
Offset: 1

Views

Author

Jonas Wallgren, Mar 17 2005

Keywords

Examples

			Square array begins:
  1,        1,     1,    1,   1,   1,   1,   1,   1, ...
  1,        3,     2,    2,   2,   2,   2,   2,   2, ...
  1,       15,     5,    4,   4,   4,   4,   4,   4, ...
  1,      105,    15,   11,  10,  10,  10,  10,  10, ...
  1,      945,    55,   23,  21,  20,  20,  20,  20, ...
  1,    10395,   232,   68,  59,  57,  56,  56,  56, ...
  1,   135135,  1161,  161, 125, 119, 117, 116, 116, ...
  1,  2027025,  6643,  488, 349, 329, 323, 321, 320, ...
  1, 34459425, 44566, 1249, 848, 760, 745, 739, 737, ...
  ...

Crossrefs

Cf. A104429-A104442. P(1, )=P(, 1) = A000012, P(_, 2) = A001147.

Extensions

More terms from Alois P. Heinz, Nov 18 2020

A332783 The number of permutations of {(n+1) 1's, (n+1) 2's, ..., (n+1) n's} with the property that k's are equally spaced for k=1..n and the interval of k+1 is less than or equal to the interval of k for k=1..n-1.

Original entry on oeis.org

1, 4, 16, 104, 484, 4848, 25104, 300336, 2335296, 27953952, 198725952, 4731323904, 33020828928, 606237831936, 8936541384192, 174694058933760, 1628654065588224, 56338295740213248, 545177455792662528, 20766878061520306176, 340162958990367645696
Offset: 1

Views

Author

Seiichi Manyama, Feb 23 2020

Keywords

Examples

			Define the interval of k as b(k).
In case of n = 1.
     |        | b(1)
-----+--------+-----
   1 | [1, 1] | [0]
In case of n = 2.
     |                    | b(1),b(2)
-----+--------------------+----------
   1 | [2, 2, 2, 1, 1, 1] | [0, 0]
   2 | [2, 1, 2, 1, 2, 1] | [1, 1]
   3 | [1, 2, 1, 2, 1, 2] | [1, 1]
   4 | [1, 1, 1, 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]

Crossrefs

Cf. A104442, A332762, A332784, A322178, A332748, A332752, A332773.

Programs

  • Ruby
    def search(a, num, d, k, n)
  if num == 0
    @cnt += 1
  else
    (k * n - k + 1).times{|i|
      if a[i] == 0
        (i + d + 1..k * n - k + 1).each{|j|
          if (k - 1) * j - (k - 2) * i < k * n
            if (1..k - 1).all?{|m| a[m * j - (m - 1) * i] == 0}
              (0..k - 1).each{|m| a[m * j - (m - 1) * i] = num}
              search(a, num - 1, j - i - 1, k, n)
              (0..k - 1).each{|m| a[m * j - (m - 1) * i] = 0}
            end
          end
        }
      end
    }
  end
end
def A(k, n)
  a = [0] * k * n
  @cnt = 0
  search(a, n, 0, k, n)
  @cnt
end
def A332783(n)
  (1..n).map{|i| A(i + 1, i)}
end
p A332783(5)

Extensions

a(9)-a(17) from Bert Dobbelaere, Mar 08 2020
a(18)-a(21) from Max Alekseyev, Sep 26 2023

A332784 The number of permutations of {n 1's, n 2's,...,n n's} with the property that b(1) >= b(2) >= ... >= b(n), where n k's are skipped by b(k) for k=1..n.

Original entry on oeis.org

5, 18, 110, 508, 4968, 25824, 305376, 2375616, 28316832, 202354752, 4771240704, 33499830528, 612464852736, 9023719675392, 176001733301760, 1649576855476224, 56693983168309248, 551579829498390528, 20888523161929138176, 342595860998544285696
Offset: 2

Views

Author

Seiichi Manyama, Feb 23 2020

Keywords

Examples

			In case of n = 2.
     |              | b(1),b(2)
-----+--------------+----------
   1 | [2, 2, 1, 1] | [0, 0]
   2 | [2, 1, 2, 1] | [1, 1]
   3 | [1, 2, 2, 1] | [2, 0]
   4 | [1, 2, 1, 2] | [1, 1]
   5 | [1, 1, 2, 2] | [0, 0]
In case of n = 3.
     |                             | b(1),b(2),b(3)
-----+-----------------------------+---------------
   1 | [3, 3, 3, 2, 2, 2, 1, 1, 1] | [0, 0, 0]
   2 | [3, 3, 3, 2, 1, 2, 1, 2, 1] | [1, 1, 0]
   3 | [3, 3, 3, 1, 2, 1, 2, 1, 2] | [1, 1, 0]
   4 | [3, 3, 3, 1, 1, 1, 2, 2, 2] | [0, 0, 0]
   5 | [3, 2, 1, 3, 2, 1, 3, 2, 1] | [2, 2, 2]
   6 | [3, 1, 2, 3, 1, 2, 3, 1, 2] | [2, 2, 2]
   7 | [1, 3, 3, 3, 1, 2, 2, 2, 1] | [3, 0, 0]
   8 | [2, 3, 1, 2, 3, 1, 2, 3, 1] | [2, 2, 2]
   9 | [1, 3, 2, 1, 3, 2, 1, 3, 2] | [2, 2, 2]
  10 | [2, 1, 3, 2, 1, 3, 2, 1, 3] | [2, 2, 2]
  11 | [1, 2, 3, 1, 2, 3, 1, 2, 3] | [2, 2, 2]
  12 | [2, 2, 2, 3, 3, 3, 1, 1, 1] | [0, 0, 0]
  13 | [1, 1, 1, 3, 3, 3, 2, 2, 2] | [0, 0, 0]
  14 | [1, 2, 2, 2, 1, 3, 3, 3, 1] | [3, 0, 0]
  15 | [2, 2, 2, 1, 1, 1, 3, 3, 3] | [0, 0, 0]
  16 | [2, 1, 2, 1, 2, 1, 3, 3, 3] | [1, 1, 0]
  17 | [1, 2, 1, 2, 1, 2, 3, 3, 3] | [1, 1, 0]
  18 | [1, 1, 1, 2, 2, 2, 3, 3, 3] | [0, 0, 0]

Crossrefs

Cf. A104442, A332762, A332783, A322178, A332748, A332752, A332773.

Formula

Conjecture: a(n) = A332783(n) + (n-1)!.

Extensions

a(9)-a(17) from Bert Dobbelaere, Mar 08 2020
a(18)-a(21) 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

A104441 Number of ways to split 1, 2, 3, ..., 8n into 8 arithmetic progressions each with n terms.

Original entry on oeis.org

1, 2027025, 6643, 488, 349, 329, 323, 321, 320, 320, 320, 320, 320, 320, 320, 320
Offset: 1

Views

Author

Jonas Wallgren, Mar 17 2005

Keywords

Crossrefs

Cf. A104429-A104440, A104442-A104443.

A090246 The largest subset of P(Z/3Z)^n that does not contain 3 collinear points.

Original entry on oeis.org

2, 4, 10, 20, 56
Offset: 1

Views

Author

Hans Havermann, Jan 23 2004

Keywords

Comments

P(Z/3Z)^n is the projective space of n dimensions over the finite field Z/3Z. This is the size of the largest subset which does not contain 3 points lying in a line.
Davis and Maclagan described a game similar to the game SET that could be played in this space using projective lines, rather than in (Z/3Z)^n using the algebraic notion of line. This sequence is the analog of A090245 for this game.
So far this sequence agrees with A104442.

Links

  • B. Davis and D. Maclagan, The Card Game SET, The Mathematical Intelligencer, Vol. 25:3 (Summer 2003), pp. 33-40.
  • B. L. Davis and D. Maclagan, The Card Game SET [From _Omar E. Pol_, Feb 21 2009]
  • Ivars Peterson, SET Math.
  • Ivars Peterson, SET Math [From _Omar E. Pol_, Feb 21 2009]

Crossrefs

Cf. A090245.

Extensions

Edited by Jack W Grahl, May 12 2009
Showing 1-6 of 6 results.

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