A248427 Circumference of the (n,n)-knight graph.

8, 14, 24, 36, 48, 64, 80, 100, 120, 144, 168, 196, 224, 256, 288, 324, 360, 400, 440, 484, 528, 576, 624, 676, 728, 784, 840, 900, 960, 1024, 1088, 1156, 1224, 1296, 1368, 1444, 1520, 1600, 1680, 1764, 1848, 1936, 2024, 2116, 2208, 2304, 2400, 2500, 2600, 2704, 2808, 2916, 3024, 3136, 3248, 3364

Offset

3,1

Comments

Colin Barker's conjectures confirmed using first Mathematica program. - Ray Chandler, Jan 14 2024

Links

Ray Chandler, Table of n, a(n) for n = 3..1000

Eric Weisstein's World of Mathematics, Graph Circumference

Eric Weisstein's World of Mathematics, Knight Graph

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Formula

From Colin Barker, Oct 07 2014: (Start)

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

a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n>8.

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

(End)

Mathematica

Table[Piecewise[{{14, n == 4}, {n^2, Mod[n, 2] == 0}, {n^2 - 1, Mod[n, 2] == 1}}], {n, 3, 50}]
CoefficientList[Series[-2x^3(x^5-2x^4+2x^3-2x^2-x+4)/((x-1)^3(x+1)), {x, 0, 50}], x] (* or *) LinearRecurrence[{2, 0, -2, 1}, {8, 14, 24, 36, 48, 64}, 50] (* Harvey P. Dale, Jan 05 2024 *)

Crossrefs

Sequence in context: A250098 A155156 A275898 * A090993 A211525 A241161

Adjacent sequences: A248424 A248425 A248426 * A248428 A248429 A248430

Keyword

nonn

Author

Eric W. Weisstein, Oct 06 2014

Status

approved