A003751 Number of spanning trees in K_5 x P_n.

125, 300125, 663552000, 1464514260125, 3232184906328125, 7133430745792512000, 15743478429512478120125, 34745849760772636969860125, 76684074678559433693601792000, 169241718069731503830237768828125, 373516395095822778319979141039280125

Offset

1,1

Comments

This is a divisibility sequence.

References

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

Links

P. Raff, 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), 129-154.

F. Faase, Counting Hamiltonian cycles in product graphs

F. Faase, Results from the counting program

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

P. Raff, Analysis of the Number of Spanning Trees of G x P_n, where G = {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}}. Contains sequence, recurrence, generating function, and more.

P. Raff, Analysis of the Number of Spanning Trees of Grid Graphs.

Index to divisibility sequences

Index entries for sequences related to trees

Index entries for linear recurrences with constant coefficients, signature (2255,-105985,105985,-2255,1).

Formula

a(n) = 2255a(n-1)- 105985a(n-2) +105985a(n-3) -2255a(n-4) +a(n-5).

a(n) = 125*(A004187(n))^4 = 125*(A049682(n))^2. [R. Guy, seqfan list, Mar 28 2009] [From R. J. Mathar, Jun 03 2009]

G.f.: -(125x(x^3+146x^2+146x+1)/(x^5-2255x^4+105985x^3-105985x^2+2255x-1)) [Paul Raff, Oct 29, 2009]

a(n) = 125*F(4n)^4/81. - R. K. Guy, Feb 24 2010

Mathematica

(125*Fibonacci[4*Range[20]]^4)/81 (* or *) LinearRecurrence[ {2255, -105985, 105985, -2255, 1}, {125, 300125, 663552000, 1464514260125, 3232184906328125}, 20] (* Harvey P. Dale, Apr 24 2013 *)

Crossrefs

Sequence in context: A050640 A161354 A318258 * A120807 A013836 A048563

Adjacent sequences: A003748 A003749 A003750 * A003752 A003753 A003754

Keyword

nonn

Author

Frans J. Faase

Extensions

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

More terms from Harvey P. Dale, Apr 24 2013

Status

approved