A003731 Number of Hamiltonian cycles in C_5 X P_n.

1, 5, 30, 160, 850, 4520, 24040, 127860, 680040, 3616880, 19236840, 102313600, 544168000, 2894227280, 15393318880, 81871340160, 435443220000, 2315960597120, 12317733383040, 65513444349760, 348441653760640, 1853231611930880, 9856649945242240, 52423856531251200

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 linear recurrences with constant coefficients, signature (6,-4,2).

Crossrefs

Column k=5 of A359855.

Programs

Magma

I:=[1,5,30,160]; [n le 4 select I[n] else 6*Self(n-1)-4*Self(n-2)+2*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Oct 14 2013

Mathematica

CoefficientList[Series[(1 - x + 4 x^2 - 2 x^3)/(1 - 6 x + 4 x^2 - 2 x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 14 2013 *)

PARI

a(n)=([0,1,0; 0,0,1; 2,-4,6]^(n-1)*[1;5;30])[1,1] \\ Charles R Greathouse IV, Jun 23 2020

Formula

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

G.f.: x*(1 - x + 4*x^2 - 2*x^3)/(1 - 6*x + 4*x^2 - 2*x^3). - Colin Barker, Sep 01 2012

Extensions

More terms from Vincenzo Librandi, Oct 14 2013