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

%I A284287 #35 Jul 04 2025 03:08:22
%S A284287 12,126720,7959229931520,10752728122249860612096000,
%T A284287 829276462388385539562198269952000000000000,
%U A284287 7969891788752886799729592752113502210704733844275200000000000000,18306383771271364475276585375748692499524930312437317320546133915243380736000000000000000000
%N A284287 Number of possible legal open chains of a set of dominoes tiles with 0 to 2n pips.
%C A284287 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).
%C A284287 The problem of finding a(2) appears in Henry Dudeney's book.
%C A284287 a(4) was calculated by Gaston Tarry in 1886.
%C A284287 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, ... .
%C A284287 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, ... .
%D A284287 Henry Ernest Dudeney, "The Fifteen Dominoes", Amusements in Mathematics, Nelson, Edinburgh 1917, pp. 209-210.
%D A284287 Martin Gardner, Mathematical Circus, Alfred A. Knopf, NY, 1979, pp. 137-142.
%D A284287 Donald E. Knuth, The Art of Computer Programming, Volume 4A, Addison-Wesley, 2011, pp. 389 and 745.
%D A284287 K. W. H. Leeflang, Domino games and domino puzzles, St. Martin's Press, New York, 1975, Chapter VIII, section 1, pp. 125-134.
%D A284287 Édouard Lucas, "La géométrie des réseaux et le problème des dominos", Récréations mathématiques, Volume 4, Gauthier-Villars, Paris, 1894, pp. 125-129.
%D A284287 Yakov Perelman, Figures for Fun, Foreign Languages Publishing House, Moscow, 1957, p. 38.
%D A284287 Miodrag S. Petković, "Poinsot's Diagram-tracing Puzzle", Famous Puzzles of Great Mathematicians, Amer. Math. Soc. (AMS), Providence RI, 2009, pp. 245-247
%D A284287 Michel 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.
%D A284287 W. W. Rouse Ball and H. S. M. Coxeter, Mathematical Recreations and Essays, Dover NY, 1987, pp. 243-254.
%H A284287 Amiram Eldar, <a href="/A284287/b284287.txt">Table of n, a(n) for n = 1..10</a>
%H A284287 Pierre Audibert, <a href="https://doi.org/10.1002/9781118557938.ch36">Enumeration of Eulerian Paths in Undirected Graphs</a>, Mathematics for Informatics and Computer Science, Wiley, 2010, Chapter 36, Section 36.4.1., "Number of domino chains", pp. 813-816.
%H A284287 Philippe Chevanne, <a href="http://mathafou.free.fr/pbm/sol233.html">Eulériens</a> (in French); <a href="https://web.archive.org/web/20170426190332/http://mathafou.free.fr:80/pbm_en/sol233.html">Eulerian circuits</a> (English translation, Wayback Machine link).
%H A284287 Henry Ernest Dudeney, <a href="https://archive.org/stream/amusementsinmath00dude#page/209/mode/1up">The Fifteen Dominoes</a>, Amusements in Mathematics, 1917.
%H A284287 Martin Gardner, <a href="https://doi.org/10.1038/scientificamerican1269-122">A handful of combinatorial problems based on dominoes</a>, Mathematical Games, Scientific American, Vol. 221, No. 6 (1969), pp. 122-127; <a href="https://www.jstor.org/stable/24964399">JSTOR link</a>.
%H A284287 Italo Ghersi, <a href="https://archive.org/details/rcin.org.pl.WA35_13877_5178_Matematica_84675/page/694/mode/2up">Matematica dilettevole e curiosa</a> (in Italian), Hoepli, Milano, 1913, p. 695.
%H A284287 Allan J. Gottlieb, <a href="https://archive.org/details/MIT-Technology-Review-1976-06/page/n67/mode/2up">Streaker at the Banquet Table</a>, Puzzle Corner, Technology Review, MIT, June 1976, pp. 64-69.
%H A284287 Donald E. Knuth, <a href="http://www.cs.utsa.edu/~wagner/knuth/fasc3a.pdf">Generating all combinations</a>, The Art of Computer Programming, Volume 4, Combinatorial Algorithms, p. 35 and 57.
%H A284287 Édouard Lucas, <a href="https://archive.org/details/recretionmatedou04lucarich">Récréations mathématiques, Vol. 4.</a>, pp. 125-129.
%H A284287 Brendan D. McKay and Robert W. Robinson, <a href="https://doi.org/10.1017/S0963548398003642">Asymptotic Enumeration of Eulerian Circuits in the Complete Graph</a>, Combinatorics, Probability and Computing, Vol. 7, No. 4 (1998), pp. 437-449; <a href="https://users.cecs.anu.edu.au/~bdm/papers/euler.pdf">author's copy</a>.
%H A284287 L. Poinsot, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k433667x/f18.item">Mémoire sur les polygones et les polyèdres</a>, J. de l'Ecole Polytechnique, Volume 4 (1810) pp. 16—49.
%H A284287 M. Reiss, <a href="https://books.google.co.il/books?id=CcumyIM6SCMC&amp;q=%22M.%20Reiss%22#v=onepage&amp;q&amp;f=false">Evaluation du nombre de combinaisons desquelles les 28 dés d'un jeu du Domino sont susceptibles d'après la règle de ce jeu</a>, Annali di Matematica Pura ed Applicata, Vol. 5.1 (1871), pp. 63-120; <a href="https://babel.hathitrust.org/cgi/pt?id=iau.31858029349713&amp;seq=71">HathiTrust link</a>.
%H A284287 G. Tarry, <a href="https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/tarry_nom.html">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</a>, Comptes Rendus Assoc. Franç. Avance. Sci. 15, part 2 (1886), pp. 49-53.
%H A284287 O. Terquem, <a href="http://www.numdam.org/item/NAM_1849_1_8__68_1">Sur les polygones et les polyèdres étoilés, polygones funiculaires</a>, Nouv. Ann. Math., Vol. 8 (1849), pp. 68-74.
%F A284287 a(n) = (n+1)*(2n+1)*n^(2n+1)*A135388(n) = (n+1)*(2n+1)*n^(2n+1)*(n-1)!^(2n+1)*A007082(n).
%e A284287 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.
%Y A284287 Cf. A000217, A007082, A135388.
%K A284287 nonn
%O A284287 1,1
%A A284287 _Amiram Eldar_, Mar 24 2017