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A302341
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Triameter of the n X n knight graph.
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0
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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
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OFFSET
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4,1
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COMMENTS
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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
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,1,-1).
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FORMULA
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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