%I #24 Apr 10 2021 06:00:42

%S 1,4,12,28,56,88,132,180,240,304,380,460,552,648,756,868,992,1120,

%T 1260,1404,1560,1720,1892,2068,2256,2448,2652,2860,3080,3304,3540,

%U 3780,4032,4288,4556,4828,5112,5400,5700,6004,6320,6640,6972,7308,7656,8008,8372

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

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

%H Michael De Vlieger, <a href="/A085622/b085622.txt">Table of n, a(n) for n = 0..10000</a>

%H Ruediger Jehn, <a href="https://arxiv.org/abs/2103.15778">Properties of Hamiltonian Circuits in Rectangular Grids</a>, arXiv:2103.15778 [math.GM], 2021.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).

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

%F From _Colin Barker_, Oct 05 2012: (Start)

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

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

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

%t 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 *)

%K nonn,easy

%O 0,2

%A _R. K. Guy_, Jul 11 2003

%E More terms from _David Wasserman_, May 30 2004