A285527 -id:A285527 - cp's OEIS frontend

A004075 Number of Skolem sequences of order n.

1, 0, 0, 6, 10, 0, 0, 504, 2656, 0, 0, 455936, 3040560, 0, 0, 1400156768, 12248982496, 0, 0, 11435578798976, 123564928167168, 0, 0, 204776117691241344, 2634563519776965376, 0, 0, 7064747252076429464064, 105435171495207196553472, 0, 0

Offset: 1

N. J. A. Sloane

Number of permutations of the multiset {1,1,2,2,...,n,n} such that the distance between the elements i equals i for every i=1,2,...,n.

Number of super perfect rhythmic tilings of [0,2n-1] with pairs. See A285698 and A285527 for the definition and tilings of triples and quadruples. - Tony Reix, Apr 25 2017

CRC Handbook of Combinatorial Designs, 1996, p. 460.

Ali Assarpour, Amotz Bar-Noy, Ou Liuo, Counting the Number of Langford Skolem Pairings, arXiv:1507.00315 [cs.DM], 2015.
S. Burrill and L. Yen, Constructing Skolem sequences via generating trees, arXiv preprint arXiv:1301.6424 [math.CO], 2013.
J. E. Miller, Langford's Problem
G. Nordh, Perfect Skolem sequences, arXiv:math/0506155 [math.CO], 2005.

Cf. A014552, A059106, A176127, A268537.

(* Program not suitable to compute a large number of terms. *)
iter[n_] := Sequence @@ Table[{x[i], {-1, 1}}, {i, 1, 2n}];
a[n_] := 1/2^(2n) Sum[Product[x[i], {i, 1, 2n}] Product[Sum[x[k] x[k+i], {k, 1, 2n-i}], {i, 1, n}], iter[n] // Evaluate];
Table[Print[a[n]]; a[n], {n, 1, 10}] (* Jean-François Alcover, Sep 29 2018, from formula in Assarpour et al. *)

For n > 1, a(n) = A059106(n)*2 because A059106 ignores reflected solutions. - Martin Fuller, Mar 08 2007

More terms (via A059106) from Martin Fuller, Mar 08 2007
Extended using results from the Assarpour et al. (2015) paper by N. J. A. Sloane, Feb 22 2016 at the suggestion of William Rex Marshall
a(28)-a(31) from Assarpour et al. (2015), added by Max Alekseyev, Sep 24 2023

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

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

Alois P. Heinz, Apr 20 2020

Differs from A331621 first at n=7.

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

Wikipedia, Partition of a set

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

Main diagonal of A360334.

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);

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 *)

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

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

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

A320392 Number of permutations of 3 indistinguishable copies of 1,...,n such that there are exactly j numbers between the first and the third copy of j and floor(j/2) numbers between the first and the second or between the second and the third copy of j.

Original entry on oeis.org

1, 1, 0, 0, 4, 4, 10, 24, 252, 410, 1998, 7798, 65188, 280582, 2281108, 10585748, 110903088

Offset: 0

Alois P. Heinz, Dec 06 2018

a(n) is even for n > 1.

			a(1) = 1: 111.
a(4) = 4: 111224234343, 111343432422, 224234343111, 343432422111.

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

Cf. A285527, A322153.

cp's OEIS Frontend

A004075 Number of Skolem sequences of order n.

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

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

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

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A320392 Number of permutations of 3 indistinguishable copies of 1,...,n such that there are exactly j numbers between the first and the third copy of j and floor(j/2) numbers between the first and the second or between the second and the third copy of j.

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