A261517 -id:A261517 - cp's OEIS frontend

A060963 Number of pairings of the first 2n positive integers so that the absolute differences of each pair are different.

1, 1, 1, 5, 29, 145, 957, 8397, 85169, 944221, 11639417, 160699437, 2430145085, 39776366397, 703161838717, 13369111112753, 271734091323897

Offset: 0

Erich Friedman, May 08 2001

Number of perfect Skolem-type sequences.

			a(3)=5 since the 5 pairings of {1, 2, 3, 4, 5, 6} are {1, 3} {2, 6} {4, 5}, {1, 5} {2, 3} {4, 6}, {1, 5} {2, 4} {3, 6}, {1, 4} {2, 6} {3, 5}, {1, 6} {2, 5} {3, 4}.

Edward Moody, Java program for calculating entries in this sequence and A322178
G. Nordh, Perfect Skolem sequences, arXiv:math/0506155 [math.CO], 2005.

Cf. A261516, A261517, A322178.

Offset corrected by Alois P. Heinz, Sep 14 2015
a(14) from Fausto A. C. Cariboni, Mar 03 2017
a(15) from Fausto A. C. Cariboni, Apr 10 2017
a(16) from Edward Moody, Feb 17 2020

A261516 Number of perfect rhythmic tilings of [0,3n-1] with triples.

Original entry on oeis.org

1, 1, 0, 0, 0, 2, 0, 18, 66, 382, 1104, 4138, 15324, 61644, 325456, 2320948, 17660110, 148271962, 1171109228, 9257051746

Offset: 0

Michel Marcus, Aug 23 2015

A perfect tiling of the line with triples consists of groups of three evenly spaced points, each group having a different common interval such that all points of the line are covered.

			For n=1, there is 1 such tiling: (0,1,2).
For n=5, there are 2 such tilings: (2,3,4), (8,10,12), (5,9,13), (1,6,11), (0,7,14) and its mirror, that have these distinct common differences: 1,2,4,5,7.

J. P. Delahaye, La musique mathématique de Tom Johnson, in Mathématiques pour le plaisir, Belin-Pour la Science, Paris, 2010.

J. P. Delahaye, La musique mathématique de Tom Johnson, Pour la Science, 325, Nov 2004, pp. 88-93.
Tom Johnson, Perfect Rhythmic Tilings, Lecture delivered at MaMuX meeting, IRCAM, January 24, 2004.
Tom Johnson, Tiling in My music, August, 2008.

Cf. A060963, A104429, A261517, A285527.

a(16)-a(17) from Alois P. Heinz, Sep 16 2015
a(18)-a(19) from Fausto A. C. Cariboni, Mar 27 2017
a(0)=1 prepended by Seiichi Manyama, Feb 21 2020

A332752 The number of permutations of {1,1,1,1,2,2,2,2,...,n,n,n,n} such that each quadruple of k's (k=1..n) is equally spaced with b(k) other elements in between, and b(1) >= b(2) >= ... >= b(n).

Original entry on oeis.org

1, 1, 4, 16, 110, 544, 5444, 32520, 385776, 3282108, 40916528, 354328560, 7200045216, 67347823160, 1182323197504, 18086875471594, 358787259407482, 4564034487662420

Offset: 0

Seiichi Manyama, Feb 22 2020

			In case of n = 1.
     |              | b(1)
-----+--------------+------
   1 | [1, 1, 1, 1] | [0] *
In case of n = 2.
     |                          | b(1),b(2)
-----+--------------------------+----------
   1 | [2, 2, 2, 2, 1, 1, 1, 1] | [0, 0]
   2 | [2, 1, 2, 1, 2, 1, 2, 1] | [1, 1]
   3 | [1, 2, 1, 2, 1, 2, 1, 2] | [1, 1]
   4 | [1, 1, 1, 1, 2, 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]
* (strongly decreasing)

Column 4 of A332762.

Cf. A104430, A261517 (strongly decreasing), A285698, A322178, A332748, A332773, A332783, A332784.

a(10)-a(17) from Max Alekseyev, Sep 27 2023

A264813 Number of permutations of 3 indistinguishable copies of 1,...,n such that the first and second copies of j are adjacent and there are exactly j numbers between the second and the third copy of j.

Original entry on oeis.org

1, 0, 1, 1, 0, 3, 6, 0, 53, 199, 0, 2908, 13699, 0, 369985, 2135430, 0, 87265700, 611286653, 0

Offset: 0

Alois P. Heinz, Nov 25 2015

a(n) = 0 for n == 1 (mod 3).

			a(0) = 1: the empty permutation.
a(2) = 1: 221121.
a(3) = 1: 223321131.
a(5) = 3: 223325534411514, 225523344531141, 552244253341131.
a(6) = 6: 221121665544336543, 225523366534411614, 225526633544361141, 446611415563322532, 552266253344631141, 665544336543221121.

Eric Weisstein's World of Mathematics, Langford's Problem
Wikipedia, Dancing Links
Wikipedia, Langford pairing

Cf. A014552, A104185, A108235, A176127, A203435, A261516, A261517, A321956.

A284757 Number of solutions to Nickerson variant of quadruples version of Langford (or Langford-Skolem) problem.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 55, 0, 0, 0, 0, 0, 0

Offset: 1

Fausto A. C. Cariboni, Apr 02 2017

How many ways are of arranging the numbers 1,1,1,1,2,2,2,2,3,3,3,3,...,n,n,n,n so that there are zero numbers between the first and second 1's, between the second and third 1's and between the third and fourth 1's; one number between the first and second 2's, between the second and third 2's and between the third and fourth 2's; ... n-1 numbers between the first and second n's, between the second and third n's and between the third and fourth n's?
An equivalent definition is A261517 with added condition that all different common intervals are <= n.
a(n) ignores reflected solutions.

Fausto A. C. Cariboni, Solutions for a(24)-a(25)
J. E. Miller, Langford's Problem.

Cf. A014552, A059106, A059108, A261517.

a(n) = 0 if (n mod 8) not in {0, 1}. - Max Alekseyev, Sep 28 2023

a(28)-a(31) from Max Alekseyev, Sep 24 2023

A285698 Number of super perfect rhythmic tilings of [0,4n-1] with quadruples.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20, 110, 0, 0, 0, 0, 0, 0

Offset: 1

Tony Reix, Apr 25 2017

A super perfect tiling of the line with quadruples consists of n groups of four evenly spaced points, each group having a different common interval such that all points of the line are covered, and such that all intervals are inferior or equal to n (thus, each interval belongs to [1..n]).

			For n = 24, there are 20 tilings.
One is: (0,3,6,9), (1,7,13,19), (2,14,26,38), (4,11,18,25), (5,29,53,77), (8,12,16,20), (10,27,44,61), (15,37,59,81), (17,33,49,65), (21,36,51,66), (22,43,64,85), (23,34,45,56), (24,47,70,93), (28,41,54,67), (30,39,48,57), (31,50,69,88), (32,46,60,74), (35,55,75,95), (40,58,76,94), (42,52,62,72), (63,71,79,87), (68,73,78,83), (80,82,84,86), (89,90,91,92)
It can also be represented as (where each number is the interval of the group the point of the line belongs to):
3 6 12 3 7 24 3 6 4 3 17 7 4 6 12 22 4 16 7 6 4 15 21 11 23 7 12 17 13 24 9 19 14 16 11 20 15 22 12 9 18 13 10 21 17 11 14 23 9 16 19 15 10 24 13 20 11 9 18 22 14 17 10 8 21 16 15 13 5 19 23 8 10 5 14 20 18 24 5 8 2 22 2 5 2 21 2 8 19 1 1 1 1 23 18 20
Another one is: (0,23,46,69), (1,25,49,73), (2,4,6,8), (3,7,11,15), (5,26,47,68), (9,14,19,24), (10,27,44,61), (12,20,28,36), (13,35,57,79), (16,34,52,70), (17,31,45,59), (18,33,48,63), (21,32,43,54), (22,42,62,82), (29,41,53,65), (30,40,50,60), (37,56,75,94), (38,51,64,77), (39,55,71,87), (58,67,76,85), (66,72,78,84), (74,81,88,95), (80,83,86,89), (90,91,92,93)
It can also be represented as:
23 24 2 4 2 21 2 4 2 5 17 4 8 22 5 4 18 14 15 5 8 11 20 23 5 24 21 17 8 12 10 14 11 15 18 22 8 19 13 16 10 12 20 11 17 14 23 21 15 24 10 13 18 12 11 16 19 22 9 14 10 17 20 15 13 12 6 9 21 23 18 16 6 24 7 19 9 13 6 22 3 7 20 3 6 9 3 16 7 3 1 1 1 1 19 7

Cf. A285527, A261517.

For n > 1, a(n) = A284757(n)*2 because A284757 ignores reflected solutions. - Fausto A. C. Cariboni, May 20 2017

a(n) = 0 if (n mod 8) not in {0, 1}, - Max Alekseyev, Sep 28 2023

a(27)-a(31) from Max Alekseyev, Sep 24 2023

cp's OEIS Frontend

A060963 Number of pairings of the first 2n positive integers so that the absolute differences of each pair are different.

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A261516 Number of perfect rhythmic tilings of [0,3n-1] with triples.

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A332752 The number of permutations of {1,1,1,1,2,2,2,2,...,n,n,n,n} such that each quadruple of k's (k=1..n) is equally spaced with b(k) other elements in between, and b(1) >= b(2) >= ... >= b(n).

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A264813 Number of permutations of 3 indistinguishable copies of 1,...,n such that the first and second copies of j are adjacent and there are exactly j numbers between the second and the third copy of j.

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A284757 Number of solutions to Nickerson variant of quadruples version of Langford (or Langford-Skolem) problem.

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A285698 Number of super perfect rhythmic tilings of [0,4n-1] with quadruples.

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