A085622 Maximal number of segments (equivalently, corners) in a rook circuit of a 2n X 2n board.

1, 4, 12, 28, 56, 88, 132, 180, 240, 304, 380, 460, 552, 648, 756, 868, 992, 1120, 1260, 1404, 1560, 1720, 1892, 2068, 2256, 2448, 2652, 2860, 3080, 3304, 3540, 3780, 4032, 4288, 4556, 4828, 5112, 5400, 5700, 6004, 6320, 6640, 6972, 7308, 7656, 8008, 8372

Offset: 0

Author

R. K. Guy, Jul 11 2003

Keywords

• nonn
• easy

References

• Problem asked by Barry Cipra arising from Problem 89 of Vaderlind, Guy & Larson, The Inquisitive Problem Solver, MAA.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• Ruediger Jehn, Properties of Hamiltonian Circuits in Rectangular Grids, arXiv:2103.15778 [math.GM], 2021.
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Programs

Mathematica

CoefficientList[Series[-(4 x^5 - 7 x^4 - 6 x^3 - 4 x^2 - 2 x - 1)/((1 - x)^3*(1 + x)), {x, 0, 46}], x] (* Michael De Vlieger, Mar 11 2021 *)

Formula

a(n) = 4n^2 - 2n if n is even and 4n^2 - 2n - 2 if n is odd and > 1.

From Colin Barker, Oct 05 2012: (Start)

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

a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n>5.

G.f.: -(4*x^5-7*x^4-6*x^3-4*x^2-2*x-1)/((1-x)^3*(1+x)). (End)

Extensions

More terms from David Wasserman, May 30 2004