A003691 Number of spanning trees with degrees 1 and 3 in K_3 X P_2n.

3, 36, 324, 2880, 25632, 228096, 2029824, 18063360, 160745472, 1430470656, 12729729024, 113281597440, 1008090611712, 8970977673216, 79832546279424, 710428191621120

Offset: 1

Author

Frans J. Faase

Keywords

• nonn
• easy

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 = 1..1000
• 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
• Index entries for sequences related to trees
• Index entries for linear recurrences with constant coefficients, signature (8,8).

Crossrefs

Cf. A057091.

Programs

Magma

i:=[3,36,324]; [n le 3 select i[n] else 8*(Self(n-1)+Self(n-2)): n in [1..16]];  // Bruno Berselli, Aug 02 2011

PARI

a(n)=([0,1; 8,8]^(n-1)*[3;36])[1,1] \\ Charles R Greathouse IV, Jun 23 2020

Formula

a(n) = 8*a(n-1) + 8*a(n-2), n>3.

From Bruno Berselli, Aug 02 2011: (Start)

G.f.: 3*x*(1+2*x)^2/(1-8*x-8*x^2).

For n>1, a(n) = 3*sqrt(3)*sqrt(2^(2*n-7))*((2+sqrt(6))^n-(2-sqrt(6))^n). (End)