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A289181 Number of 6-cycles in the n X n knight graph.

%I A289181 #13 Feb 16 2025 08:33:48
%S A289181 0,0,0,20,164,616,1348,2352,3628,5176,6996,9088,11452,14088,16996,
%T A289181 20176,23628,27352,31348,35616,40156,44968,50052,55408,61036,66936,
%U A289181 73108,79552,86268,93256,100516,108048,115852,123928,132276,140896,149788,158952,168388,178096
%N A289181 Number of 6-cycles in the n X n knight graph.
%H A289181 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>
%H A289181 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KnightGraph.html">Knight Graph</a>
%H A289181 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3, -3, 1).
%F A289181 For n > 5, a(n) = 4 (34 n^2 - 259 n + 484).
%F A289181 G.f.: (4*x^4*(-5 - 26*x - 46*x^2 + 7*x^3 + 2*x^4))/(-1 + x)^3.
%t A289181 Table[Length[FindCycle[KnightTourGraph[n, n], {6}, All]], {n, 20}]
%t A289181 Table[Piecewise[{{0, n < 4}, {20, n == 4}, {164, n == 5}}, 4 (34 n^2 - 259 n + 484)], {n, 20}]
%t A289181 Join[{0, 0, 0, 20, 164}, LinearRecurrence[{3, -3, 1}, {1036, 408, 52}, {6, 20}]]
%t A289181 CoefficientList[Series[(4 x^3 (-5 - 26 x - 46 x^2 + 7 x^3 + 2 x^4))/(-1 + x)^3, {x, 0, 20}], x]
%o A289181 (PARI) a(n)=4*if(n>5,34*n^2 - 259*n + 484, max(36*n-139, 0)) \\ _Charles R Greathouse IV_, Oct 21 2022
%Y A289181 Cf. A163433 (4-cycles).
%K A289181 nonn,easy
%O A289181 1,4
%A A289181 _Eric W. Weisstein_, Jun 27 2017