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%I #6 Oct 12 2012 14:55:30
%S -1,-1,-1,-2,0,1,3,3,-1,-1,5,0,-5,0,1,-8,-8,6,6,-1,-1,-13,0,19,0,-8,0,
%T 1,21,21,-25,-25,9,9,-1,-1,34,0,-65,0,42,0,-11,0,1,-55,-55,90,90,-51,
%U -51,12,12,-1,-1,-89,0,210,0,-183,0,74,0,-14,0,1
%N Alternating sign center tridiagonal matrices as triangular sequences: m(n,m,d)=If[ n == m, (-1)^n, If[n == m - 1 || n == m + 1, -1, 0]].
%C The scalar change effect between first elements of the matrix tridiagonals and their recursive polynomial counter parts reminded me of pseudoscalar results in tensors ( sign changes of scalars). Determinant sequence is Fibonacci: Table[Det[M[d]], {d, 1, 10}] {-1, -2, 3, 5, -8, -13, 21, 34, -55, -89} Matrices: 1 X 1 {{-1}}, 2 X 2 {{-1, -1}, {-1, 1}}, 3 X 3 {{-1, -1,0}, {-1, 1, -1}, {0, -1, -1}}, 4 X 4 {{-1, -1, 0, 0}, {-1, 1, -1, 0}, {0, -1, -1, -1}, {0, 0, -1, 1}}, 5 X 5 {{-1, -1, 0, 0, 0}. {-1, 1, -1, 0, 0}, {0, -1, -1, -1, 0}, {0, 0, -1, 1, -1}, {0, 0, 0, -1, -1}}, 6 X 6 {{-1, -1, 0, 0, 0, 0}, {-1, 1, -1, 0, 0, 0}, {0, -1, -1, -1, 0, 0}, {0, 0, -1, 1, -1, 0}, {0, 0, 0, -1, -1, -1}, {0, 0, 0, 0, -1, 1}}
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Pseudoscalar.html">Pseudoscalar</a>
%F m(n,m,d)=If[ n == m, (-1)^n, If[n == m - 1 || n == m + 1, -1, 0]]
%e Triangular sequence:
%e {-1}},
%e {-1, -1},
%e {-2, 0, 1},
%e {3, 3, -1, -1},
%e {5, 0, -5, 0, 1},
%e {-8, -8, 6, 6, -1, -1},
%e {-13, 0, 19, 0, -8, 0, 1},
%e {21, 21, -25, -25, 9, 9, -1, -1},
%e {34, 0, -65, 0, 42, 0, -11, 0, 1},
%e {-55, -55, 90, 90, -51, -51, 12, 12, -1, -1},
%e {-89, 0, 210, 0, -183, 0, 74, 0, -14, 0, 1}
%t T[n_, m_, d_] := If[ n == m, (-1)^n, If[n == m - 1 || n == 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 tabl,uned,sign
%O 1,4
%A _Roger L. Bagula_ and _Gary W. Adamson_, Nov 01 2006
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