|
%I #24 Mar 02 2023 22:59:17
%S 0,2,42,400,2840,17376,97440,516608,2634624,13058560,63320576,
%T 301707264,1417009152,6575120384,30195425280,137430827008,
%U 620604391424,2783097520128,12403773407232,54975376916480,242441862512640,1064326263734272,4653131038195712,20266193591992320
%N Number of ways to place two dimers on an n-cube.
%C Equivalently, a(n) is the number of 2-matchings in the n-hypercube graph. A 2-matching is a pair of edges that do not share a vertex.
%H Krishnan Balasubramanian, <a href="https://doi.org/10.3390/sym15020557">Topological Indices, Graph Spectra, Entropies, Laplacians, and Matching Polynomials of n-Dimensional Hypercubes</a>, Symmetry. 2023; 15(2):557.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HypercubeGraph.html">Hypercube Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Matching-GeneratingPolynomial.html">Matching-Generating Polynomial</a>.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (18,-132,504,-1056,1152,-512).
%F a(n) = A192437(n, 2^(n-1)-2) for n > 1.
%F From _Andrew Howroyd_, Feb 20 2023: (Start)
%F a(n) = 2^(n-2)*n*(1 - 2*n + n*2^(n-1)).
%F a(n) = 18*a(n-1) - 132*a(n-2) + 504*a(n-3) - 1056*a(n-4) + 1152*a(n-5) - 512*a(n-6) for n > 6.
%F G.f.: 2*x^2*(1 + 3*x - 46*x^2 + 88*x^3)/((1 - 2*x)*(1 - 4*x))^3.
%F (End)
%e The a(2) = 2 2-matchings are:
%e o---o o o
%e | |
%e o---o o o
%o (PARI) a(n) = 2^(n-2)*n*(1 - 2*n + n*2^(n-1)) \\ _Andrew Howroyd_, Feb 20 2023
%Y Column k=2 of A302235.
%Y Third from last terms in rows of A192437.
%Y Cf. A001787 (1-matchings), A045310 (matchings).
%K nonn,easy
%O 1,2
%A _Krishnan Balasubramanian_, Feb 20 2023
%E Terms a(13) and beyond from _Andrew Howroyd_, Feb 20 2023
| 