A007786 Number of nonintersecting rook paths joining opposite corners of 4 X n board.

1, 8, 38, 184, 976, 5382, 29739, 163496, 896476, 4913258, 26932712, 147657866, 809563548, 4438573234, 24335048679, 133419610132, 731487691902, 4010463268476, 21987818897998, 120550710615560, 660932932108467

Offset: 1

Author

Heiner Marxen

Keywords

• nonn
• easy
• nice
• walk

References

• Netnews group rec.puzzles, Frequently Asked Questions (FAQ) file. (Science Section).

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• F. Faase, Counting Hamiltonian cycles in product graphs
• F. Faase, Results from the counting program
• F. Faase, Rook path problem
• D. G. Radcliffe, N. J. A. Sloane, C. Cole, J. Gillogly, & D. Dodson, Emails, 1994
• Index entries for linear recurrences with constant coefficients, signature (12,-54,124,-133,-16,175,-94,-69,40,12,-4,-1).

Crossrefs

Row 4 of A064298.

Cf. A006192, A007764, A007787.

Programs

Mathematica

LinearRecurrence[{12,-54,124,-133,-16,175,-94,-69,40,12,-4,-1},{1,8,38,184,976,5382,29739,163496,896476,4913258,26932712,147657866},30] (* Harvey P. Dale, Jun 27 2012 *)

Formula

a(n) = 12*a(n - 1) - 54*a(n - 2) + 124*a(n - 3) - 133*a(n - 4) - 16*a(n - 5) + 175*a(n - 6) - 94*a(n - 7) - 69*a(n - 8) + 40*a(n - 9) + 12*a(n - 10) - 4*a(n - 11) - a(n - 12). - Vladeta Jovovic, Mar 20 2000

G.f.: x*(x^10-15*x^8+6*x^7+50*x^6-26*x^5-39*x^4+36*x^3-4*x^2-4*x+1) / ((x^6+2*x^5-9*x^4-5*x^3+15*x^2-8*x+1)*(x^6+2*x^5-7*x^4-3*x^3+7*x^2-4*x+1)). [Colin Barker, Nov 24 2012]

Extensions

More terms from Vladeta Jovovic, Mar 20 2000