A302341 Triameter of the n X n knight graph.

12, 12, 12, 12, 16, 16, 18, 20, 22, 22, 24, 26, 28, 30, 32, 32, 36, 36, 38, 40, 42, 42, 44, 46, 48, 50, 52, 52, 56, 56, 58, 60, 62, 62, 64, 66, 68, 70, 72, 72, 76, 76, 78, 80, 82, 82, 84, 86, 88, 90, 92, 92, 96, 96, 98, 100, 102, 102, 104, 106, 108, 110, 112, 112

Offset

4,1

Comments

In many cases, with three knights in different corners the sum of the distances between them will equal the triameter (specifically when n mod 12 is 0, 2, 3, 4, 7, 8, 11). In other cases, moving one of pieces by one square will increase the total distance by 2. It has not been proven that these constructions are optimal. - Andrew Howroyd, Apr 06 2018

Links

Table of n, a(n) for n=4..67.

Eric Weisstein's World of Mathematics, Graph Triameter

Eric Weisstein's World of Mathematics, Knight Graph

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

Formula

a(n) = 20 + a(n-12) for n > 25. - Andrew Howroyd, Apr 06 2018

a(n) = 20*k + a(n-12*k) for n > 12*k+5. - Eric W. Weisstein, Apr 15 2018

a(n) = a(n-1) + a(n-12) - a(n-13) for n > 19. - Eric W. Weisstein, Apr 14 2018

G.f.: 2*x^4*(6 + 2*x^4 + x^6 + x^7 + x^8 + x^10 + x^11 - 5*x^12 + x^13 + x^14)/((-1 + x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11)). - Eric W. Weisstein, Apr 14 2018

Mathematica

Join[{12, 12}, Table[1/6 (8 + 2 (-1)^n + 10 n + Cos[n Pi/3] (1 - 2 (-1)^n Cos[n Pi/2]) + Cos[2 n Pi/3] - Sqrt[3] Sin[n Pi/3] + 2 Cos[n Pi/2] (1 + (-1)^n Sqrt[3] Sin[n Pi/3]) - (5 Sin[2 n Pi/3])/Sqrt[3]), {n, 6, 67}]] (* Eric W. Weisstein, Apr 14 2018 *)
Join[{12, 12}, LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1}, {12, 12, 16, 16, 18, 20, 22, 22, 24, 26, 28, 30, 32}, 62]] (* Eric W. Weisstein, Apr 14 2018 *)
CoefficientList[Series[2 (6 + 2 x^4 + x^6 + x^7 + x^8 + x^10 + x^11 - 5 x^12 + x^13 + x^14)/((-1 + x)^2 (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11)), {x, 0, 63}], x] (* Eric W. Weisstein, Apr 14 2018 *)

Crossrefs

Sequence in context: A343194 A123896 A122878 * A064162 A028996 A113595

Adjacent sequences: A302338 A302339 A302340 * A302342 A302343 A302344

Keyword

nonn

Author

Eric W. Weisstein, Apr 05 2018

Extensions

a(31)-a(50) from Andrew Howroyd, Apr 06 2018

a(51)-a(67) from Giovanni Resta, Apr 09 2018

Status

approved