cp's OEIS Frontend

A242856 Number of 2-matchings of the n X n grid graph.

%I A242856 #48 Jun 04 2024 17:14:10
%S A242856 2,44,224,686,1622,3272,5924,9914,15626,23492,33992,47654,65054,86816,
%T A242856 113612,146162,185234,231644,286256,349982,423782,508664,605684,
%U A242856 715946,840602,980852,1137944,1313174,1507886,1723472,1961372,2223074,2510114,2824076,3166592
%N A242856 Number of 2-matchings of the n X n grid graph.
%C A242856 Number of ways two dominoes can be placed on an n X n chessboard.
%H A242856 Alois P. Heinz, <a href="/A242856/b242856.txt">Table of n, a(n) for n = 2..1000</a>
%H A242856 Ralf Stephan, <a href="/A242856/a242856_1.pdf">In how many ways can we place two dominoes on the n x n chessboard?</a> Proof of formula.
%H A242856 Ralf Stephan, <a href="/A242856/a242856.pdf">Two dominoes on the 3x3 chessboard</a>, illustration of a(3)=44.
%H A242856 <a href="/index/Do#domino">Index entries for sequences related to dominoes</a>.
%H A242856 <a href="/index/Mat#matchings">Index entries for sequences related to matchings</a>.
%H A242856 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A242856 a(n) = 2*n^4 - 4*n^3 - 5*n^2 + 13*n - 4.
%F A242856 G.f.: -2*x^2*(x^4-7*x^3+12*x^2+17*x+1) / (x-1)^5. - _Colin Barker_, Jun 26 2014
%F A242856 a(n + 1) = (1/2)*A046092(n)*(A046092(n) - 1) - A016742(n) - A054000(n). - _Nicolas Bělohoubek_, May 15 2024
%F A242856 E.g.f.: 4 - 2*x + exp(x)*(2*x^4 + 8*x^3 - 3*x^2 + 6*x - 4). - _Stefano Spezia_, Jun 04 2024
%t A242856 LinearRecurrence[{5, -10, 10, -5, 1}, {2, 44, 224, 686, 1622}, 50] (* _Paolo Xausa_, May 20 2024 *)
%o A242856 (Sage)
%o A242856 def a(n):
%o A242856     G = Graph(graphs.Grid2dGraph(n,n))
%o A242856     G.relabel()
%o A242856     return G.matching_polynomial()[n^2-4]
%o A242856 (PARI) Vec(-2*x^2*(x^4-7*x^3+12*x^2+17*x+1)/(x-1)^5 + O(x^100)) \\ _Colin Barker_, Jun 26 2014
%Y A242856 Second column of A242861. Cf. A016742, A046092, A054000, A210662.
%K A242856 nonn,easy
%O A242856 2,1
%A A242856 _Ralf Stephan_, May 24 2014
%E A242856 a(7)-a(36) from _Alois P. Heinz_, Jun 01 2014