

A003696


Number of spanning trees in P_4 X P_n.


4



1, 56, 2415, 100352, 4140081, 170537640, 7022359583, 289143013376, 11905151192865, 490179860527896, 20182531537581071, 830989874753525760, 34214941811800329425, 1408756312731277540744
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OFFSET

1,2


COMMENTS

Also number of domino tilings of the 7 X (2n1) rectangle with upper left corner removed.  Alois P. Heinz, Apr 14 2011
A linear divisibility sequence of order 8; a(n) divides a(m) whenever n divides m. It is the product of a 2ndorder Lucas sequence and a 4th order linear divisibility sequence.  Peter Bala, Apr 27 2014


REFERENCES

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129154.


LINKS

T. D. Noe, Table of n, a(n) for n = 1..200
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), 129154.
F. Faase, Counting Hamilton cycles in product graphs
F. Faase, Results from the counting program
P. Raff, Spanning Trees in Grid Graphs.
P. Raff, Analysis of the Number of Spanning Trees of P_4 x P_n. Contains sequence, recurrence, generating function, and more.
Index entries for sequences related to trees


FORMULA

a(1) = 1,
a(2) = 56,
a(3) = 2415,
a(4) = 100352,
a(5) = 4140081,
a(6) = 170537640,
a(7) = 7022359583,
a(8) = 289143013376 and
a(n) = 56a(n1)  672a(n2) + 2632a(n3)  4094a(n4) + 2632a(n5)  672a(n6) + 56a(n7)  a(n8).
G.f.: x(x^649x^4+112x^349x^2+1) / (x^856x^7 +672x^62632x^5 +4094x^4 2632x^3 +672x^256x+1). [From Paul Raff, Mar 06 2009]
From Peter Bala, Apr 27 2014: (Start)
a(n) = Resultant( U(3,(x4)/2),U(n1,x/2) ), where U(n,x) denotes the Chebyshev polynomial of the second kind. The polynomial U(3,(x4)/2) = x^3  12*x^2 + 46*x  56 (see A159764) has zeros z_1 = 4, z_2 = 4 + sqrt(2) and z_3 = 4  sqrt(2). Hence a(n) = U(n1,2)*U(n1,1/2*(4 + sqrt(2)))*U(n1,1/2*(4  sqrt(2))).
a(n) = A001353(n)*A161158(n1). (End)


MAPLE

seq(resultant(simplify(ChebyshevU(3, (x4)*(1/2))), simplify(ChebyshevU(n1, (1/2)*x)), x), n = 1 .. 14);  Peter Bala, Apr 27 2014


CROSSREFS

A row of A116469.  N. J. A. Sloane, May 27 2012
Bisection of A189004.  Alois P. Heinz, Sep 20 2012
A001353, A161158, A159764.
Sequence in context: A221398 A124101 A198948 * A199709 A205227 A224176
Adjacent sequences: A003693 A003694 A003695 * A003697 A003698 A003699


KEYWORD

nonn,easy


AUTHOR

Frans J. Faase


EXTENSIONS

Added recurrence from Faase's web page.  N. J. A. Sloane, Feb 03 2009


STATUS

approved



