A003757 Number of perfect matchings (or domino tilings) in D_4 X P_(n-1).

0, 1, 1, 6, 13, 49, 132, 433, 1261, 3942, 11809, 36289, 109824, 335425, 1018849, 3104934, 9443629, 28756657, 87504516, 266383153, 810723277, 2467770054, 7510988353, 22861948801, 69584925696, 211799836801, 644660351425

Offset

0,4

Comments

Here D_4 is the graph on 4 vertices with edges (1,2), (1,3), (2,3), (1.4): a triangular kite with a tail.

This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m). - T. D. Noe, Dec 22 2008

This is the case P1 = 1, P2 = -8, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 31 2014

References

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..160

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.

F. Faase, Counting Hamiltonian cycles in product graphs

F. Faase, Results from the counting program

F. J. Faase, Results from the counting program

Paul Raff, Spanning Trees in Grid Graphs, arXiv:0809.2551 [math.CO], 2008.

H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences, Integers, Volume 12A (2012) The John Selfridge Memorial Volume

Index to divisibility sequences

Index entries for sequences related to dominoes

Index entries for linear recurrences with constant coefficients, signature (1,6,1,-1).

Formula

a(n) = a(n-1) + 6a(n-2) + a(n-3) - a(n-4), n>4.

G.f.: x(1-x^2)/(1-x-6x^2-x^3+x^4). [T. D. Noe, Dec 22 2008]

From Peter Bala, Mar 31 2014: (Start)

a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), where alpha = (1 + sqrt(33))/4 and beta = (1 - sqrt(33))/4 and T(n,x) denotes the Chebyshev polynomial of the first kind.

a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 2; 1, 1/2].

a(n) = U(n-1,i*(1 + sqrt(3))/sqrt(8))*U(n-1,i*(1 - sqrt(3))/sqrt(8)), where U(n,x) denotes the Chebyshev polynomial of the second kind.

See the remarks in A100047 for the general connection between Chebyshev polynomials and 4th-order linear divisibility sequences. (End)

Mathematica

CoefficientList[Series[x(1-x^2)/(1-x-6x^2-x^3+x^4), {x, 0, 30}], x] (* T. D. Noe, Dec 22 2008 *)
LinearRecurrence[{1, 6, 1, -1}, {0, 1, 1, 6}, 40] (* Harvey P. Dale, Sep 23 2011 *)

Prog

(Magma) I:=[0, 1, 1, 6]; [n le 4 select I[n] else Self(n-1)+6*Self(n-2)+Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Sep 24 2011

Crossrefs

Sequence in context: A131188 A247939 A203977 * A187985 A320043 A296619

Adjacent sequences: A003754 A003755 A003756 * A003758 A003759 A003760

Keyword

nonn

Author

Frans J. Faase

Extensions

Offset and name changed by T. D. Noe, Dec 22 2008

0 and 1 prepended by T. D. Noe, Dec 22 2008

Status

approved