A038758 - OEIS

%I #25 Sep 08 2022 08:44:53

%S 16,281,1785,7175,22015,56406,126966,259170,490050,871255,1472471,

%T 2385201,3726905,5645500,8324220,11986836,16903236,23395365,31843525,

%U 42693035,56461251,73744946,95228050,121689750,154012950,193193091

%N Number of ways of covering a 2n X 2n lattice by 2n^2 dominoes with exactly 4 horizontal (or vertical) dominoes.

%H Vincenzo Librandi, <a href="/A038758/b038758.txt">Table of n, a(n) for n = 2..1000</a>

%H M. E. Fisher, <a href="http://dx.doi.org/10.1103/PhysRev.124.1664">Statistical mechanics of dimers on a plane lattice</a>, Physical Review, 124 (1961), 1664-1672.

%H P. W. Kasteleyn, <a href="http://dx.doi.org/10.1016/0031-8914(61)90063-5">The Statistics of Dimers on a Lattice</a>, Physica, 27 (1961), 1209-1225.

%H <a href="/index/Do#domino">Index entries for sequences related to dominoes</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F a(n) = (1/24)*n*(n-1)*(n+1)*(12*n^3-11*n-10).

%F G.f.: x^2*(16+169*x+154*x^2+21*x^3)/(1-x)^7. [_Colin Barker_, Jun 26 2012]

%e a(3) = 281 because we have 281 ways to cover a 4 X 4 lattice with exactly 4 horizontal dominoes and exactly 14 vertical dominoes.

%t CoefficientList[Series[(16 + 169 x + 154 x^2 + 21 x^3)/(1 - x)^7, {x, 0, 30}], x] (* _Vincenzo Librandi_, Oct 22 2013 *)

%o (Magma) [(1/24)*n*(n-1)*(n+1)*(12*n^3-11*n-10): n in [2..30]]; // _Vincenzo Librandi_, Oct 22 2013

%Y Cf. A004003, A002414, A054344.

%K nonn,easy

%O 2,1

%A Yong Kong (ykong(AT)curagen.com), May 06 2000

%E More terms from _James A. Sellers_, May 10 2000