A284287 Number of possible legal open chains of a set of dominoes tiles with 0 to 2n pips.

12, 126720, 7959229931520, 10752728122249860612096000, 829276462388385539562198269952000000000000, 7969891788752886799729592752113502210704733844275200000000000000, 18306383771271364475276585375748692499524930312437317320546133915243380736000000000000000000

Offset

1,1

Comments

a(3) corresponds to the standard double-six set of 28 tiles. The question for its value was asked by Louis Poinsot in 1809 and by Orly Terquem in 1849 and was first calculated by Michel Reiss in 1859 (published in 1871).

The problem of finding a(2) appears in Henry Dudeney's book.

a(4) was calculated by Gaston Tarry in 1886.

The number of legally closed chains is a(n)/((n+1)*(2n+1)) = n^(2n+1) * A135388(n) (i.e., divided by the number of tiles in the set, A000217(2n+1)) = 2, 8448, 284258211840, 238949513827774680268800, ...

If reverse order is not counted, the number of open chains is a(n)/2 = 6, 63360, 3979614965760, 5376364061124930306048000, ... and the number of closed chains is a(n)/(2*(n+1)*(2n+1)) = 1, 4224, 142129105920, 119474756913887340134400, ...

References

H. E. Dudeney, "The Fifteen Dominoes", Amusements in Mathematics, Nelson, Edinburgh 1917, pp. 209-210.

M. Gardner, Mathematical Circus, Alfred A. Knopf, NY, 1979, pp. 137-142.

D. E. Knuth, The Art of Computer Programming, Volume 4A, Addison-Wesley, 2011, pp. 389 and 745.

É. Lucas, "La géométrie des réseaux et le problème des dominos", Récréations mathématiques, Volume 4, Gauthier-Villars, Paris, 1894, pp. 125-129.

M. Petkovic, "Poinsot's Diagram-tracing Puzzle", Famous Puzzles of Great Mathematicians, Amer. Math. Soc. (AMS), Providence RI, 2009, pp. 245-247

M. Reiss, Evaluation du nombre de combinaisons desquelles les 28 dés d'un jeu du Domino sont susceptibles d'après la règle de ce jeu, Annali di Matematica Pura ed Applicata, Vol. 5.1 (1871), pp. 63-120.

W. W. Rouse Ball & H. S. M. Coxeter, Mathematical Recreations and Essays, Dover NY, 1987, pp. 243-254.

Links

Table of n, a(n) for n=1..7.

Philippe Chevanne, Eulerian circuits

Henry Ernest Dudeney, The Fifteen Dominoes, Amusements in Mathematics, 1917.

Donald E. Knuth, Generating all combinations, The Art of Computer Programming, Volume 4, Combinatorial Algorithms, p. 35 and 57.

Édouard Lucas, Récréations mathématiques, Vol. 4., pp. 125-129.

Brendan D. McKay and Robert W. Robinson, Asymptotic enumeration of eulerian circuits in the complete graph, Combinatorics, Probability and Computing, Vol. 7 No. 4 (1998), pp. 437-449.

L. Poinsot, Mémoire sur les polygones et les polyèdres, J. de l'Ecole Polytechnique, Volume 4 (1810) pp. 16—49.

M. Reiss, Evaluation du nombre de combinaisons desquelles les 28 dés d'un jeu du Domino sont susceptibles d'après la règle de ce jeu, Annali di Matematica Pura ed Applicata, Vol. 5.1 (1871), pp. 63-120.

G. Tarry, Géométrie de situation: Nombre de manières distinctes de parcourir en une seule course toutes les allées d'un labyrinthe rentrant, en ne passant qu'une seule fois par chacune des allées, Comptes Rendus Assoc. Franç. Avance. Sci. 15, part 2 (1886), pp. 49-53.

O. Terquem, Sur les polygones et les polyèdres étoilés, polygones funiculaires, Nouv. Ann. Math., Vol. 8 (1849), pp. 68-74.

Formula

a(n) = (n+1)*(2n+1)*n^(2n+1)*A135388(n) = (n+1)*(2n+1)*n^(2n+1)*(n-1)!^(2n+1)*A007082(n).

Example

For n=1 there is 1 basic chain of 6 tiles: (0|0)(0|1)(1|1)(1|2)(2|2)(2|0). There are 6 cyclic permutations and a 2nd version for each, in a reverse order, so a(1) = 1 * 6 * 2 = 12.

Crossrefs

Cf. A007082, A135388.

Sequence in context: A369523 A328992 A069048 * A100246 A055323 A368686

Adjacent sequences: A284284 A284285 A284286 * A284288 A284289 A284290

Keyword

nonn

Author

Amiram Eldar, Mar 24 2017

Status

approved