A003733 Number of spanning trees in C_5 X P_n.

5, 1805, 508805, 140503005, 38720000000, 10668237057005, 2939274449134805, 809816405722655805, 223117116976138566005, 61472262298219520000000, 16936571572967914651674005, 4666290873812984282155907805, 1285636259054921313298518442805

Offset: 1

Author

Frans J. Faase

Keywords

• nonn

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. [Added by _Paul Raff_, Oct 30 2009]
• P. Raff, Analysis of the Number of Spanning Trees of C_5 x P_n. Contains sequence, recurrence, generating function, and more. [Added by _Paul Raff_, Oct 30 2009] [broken link]
• P. Raff, Analysis of the Number of Spanning Trees of Grid Graphs. [Added by _Paul Raff_, Oct 30 2009] [broken link]
• Index entries for sequences related to trees
• Index entries for linear recurrences with constant coefficients, signature (319, -12441, 128319, -408001, 408801, -128319, 12441, -319, 1).

Crossrefs

Cf. A143699.

Programs

Maple

a:= n-> (Matrix(1,9, (i,j)-> [0, 5, 1805, 508805, 140503005][1+abs(j-5)]). Matrix(9, (i,j)-> if (i=j-1) then 1 elif j=1 then -[408001, 128319, 12441, 319, 1][1/2+abs(i-9/2)] *(-1)^i else 0 fi)^n)[1, 5]: seq(a(n), n=1..20); # Alois P. Heinz, Mar 28 2009

Mathematica

a[n_] := (16/41)*Sinh[n*ArcCosh[(-9 - Sqrt[5])/4]]^2*Sinh[n*ArcCosh[(-9 + Sqrt[5])/4]]^2 // Round; Array[a, 20] (* Jean-François Alcover, Jan 31 2016, after Peter Bala in A143699 *)

Formula

a(n) = 319*a(n-1) - 12441*a(n-2) + 128319*a(n-3) - 408001*a(n-4) + 408001*a(n-5) - 128319*a(n-6) + 12441*a(n-7) - 319*a(n-8) + a(n-9). [Modified by Paul Raff, Oct 30 2009]

G.f.: -5*x *(1+x) *(x^6+41*x^5-998*x^4+2722*x^3-998*x^2+41*x+1) / ( (x-1)*(x^4-279*x^3+961*x^2-279*x+1) *(x^4-39*x^3+281*x^2-39*x+1) ).

a(n) = 5 * (A143699(n))^2. - R. K. Guy, Mar 11 2010

Extensions

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