A006864 Number of Hamiltonian cycles in P_4 X P_n. (Formerly M1603)

0, 1, 2, 6, 14, 37, 92, 236, 596, 1517, 3846, 9770, 24794, 62953, 159800, 405688, 1029864, 2614457, 6637066, 16849006, 42773094, 108584525, 275654292, 699780452, 1776473532, 4509783909, 11448608270, 29063617746, 73781357746, 187302518353, 475489124976

Offset

1,3

Comments

Wazir tours on a 4 X n grid. There are two closed loops for a 4x4 board, appearing as an H and a C, for example. - Ed Pegg Jr, Sep 07 2010

References

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

Kwong, Y. H. H.; Enumeration of Hamiltonian cycles in P_4 X P_n and P_5 X P_n. Ars Combin. 33 (1992), 87-96.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=1..31.

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

C. Flye Sainte-Marie, Manières différentes de tracer une route fermée ..., L'Intermédiaire des Mathématiciens, vol. 11 (1904), pp. 86-88 (in French).

George Jelliss, Wazir Wanderings

R. Tosic, O. Bodroza, Y. H. Harris Kwong, and H. Joseph Straight, On the number of Hamiltonian cycles of P4 X Pn, Indian J. Pure Appl. Math. 21 (5) (1990), 403-409.

Index entries for linear recurrences with constant coefficients, signature (2,2,-2,1).

Formula

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

G.f.: x^2/(1-2x-2x^2+2x^3-x^4). - R. J. Mathar, Dec 16 2008

a(n)=sum ( sum ( binomial(k,j) * sum (binomial(j, i-j)*2^j *binomial(k-j,n-i-3*(k-j))*(-2)^(4*(k-j)-(n-i)), i,j,n-k+j) , j,0,k) , k,1,n ), n>0. - Vladimir Kruchinin, Aug 04 2010

a(n) = Sum_{k=1..n-1} A181688(k). - Kevin McShane, Aug 04 2019

Prog

(Maxima) a(n):=sum ( sum ( binomial(k, j) *sum (binomial(j, i-j)*2^j *binomial(k-j, n-i-3*(k-j))*(-2)^(4*(k-j)-(n-i)), i, j, n-k+j) , j, 0, k) , k, 1, n ); /* Vladimir Kruchinin, Aug 04 2010 */

Crossrefs

Row/column 4 of A321172.

Sequence in context: A248113 A339985 A026598 * A217420 A071636 A263758

Adjacent sequences: A006861 A006862 A006863 * A006865 A006866 A006867

Keyword

easy,nonn

Author

kwong(AT)cs.fredonia.edu (Harris Kwong), N. J. A. Sloane, Simon Plouffe and Frans J. Faase

Status

approved