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%I #7 Aug 09 2015 01:14:01
%S 3,3,-1,8,-6,1,20,-24,9,-1,45,-84,50,-12,1,125,-275,225,-85,15,-1,320,
%T -864,900,-468,129,-18,1,845,-2639,3339,-2219,840,-182,21,-1,2205,
%U -7896,11756,-9528,4610,-1368,244,-24,1,5780,-23256,39825,-38121,22518,-8532,2079,-315,27,-1,15125,-67650,130975
%N 3 center "cycle" matrices as a triangular sequence based on the triangular model: m(n,m,3)={{3, 1, 1}, {1, 3, 1}, {1, 1, 3}} m(n,m,d)=If[ n == m, 3, If[n == m - 1 || n ==m + 1, 1, If[(n == 1 && m == d) || (n == d && m == 1), 1, 0]]].
%C Matrices: 1 X 1 {{3}}, 2 X 2 {{3, 1}, {1, 3}}, 3 X 3 {{3, 1,1}, {1, 3, 1}, {1, 1, 3}}, 4 X 4 {{3, 1, 0, 1}, {1, 3, 1, 0}, {0, 1, 3, 1}, {1, 0, 1, 3}}, 5 X 5 {{3, 1, 0, 0, 1}, {1, 3, 1, 0, 0}, {0, 1, 3, 1, 0}, {0, 0, 1, 3, 1}, {1, 0, 0, 1, 3}} Large Root: Table[x /. NSolve[Det[M[d] - x*IdentityMatrix[d]] == 0, x][[d]], {d, 1, 10}] {3., 4., 5., 5., 5., 5., 5., 5., 5., 5.}
%F m(n,m,d)=If[ n == m, 3, If[n == m - 1 || n ==m + 1, 1, If[(n == 1 && m == d) || (n == d && m == 1), 1, 0]]]
%e Triangle begins:
%e {3},
%e {3, -1},
%e {8, -6, 1},
%e {20, -24, 9, -1},
%e {45, -84, 50, -12, 1},
%e {125, -275,225, -85, 15, -1}
%t T[n_, m_, d_] := If[ n == m, 3, If[n == m - 1 || n == m + 1, 1, If[(n == 1 && m == d) || (n == d && m == 1), 1, 0]]] M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}] Table[M[d], {d, 1, 10}] Table[Det[M[d]], {d, 1, 10}] Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}] a = Join[M[1], Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]] Flatten[a] MatrixForm[a]
%K uned,tabl,sign
%O 1,1
%A _Gary W. Adamson_ and _Roger L. Bagula_, Nov 04 2006
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