A124031 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]].

-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, 1, 21, 21, -25, -25, 9, 9, -1, -1, 34, 0, -65, 0, 42, 0, -11, 0, 1, -55, -55, 90, 90, -51, -51, 12, 12, -1, -1, -89, 0, 210, 0, -183, 0, 74, 0, -14, 0, 1

Offset

1,4

Comments

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}}

Links

Table of n, a(n) for n=1..66.

Eric Weisstein's World of Mathematics, Pseudoscalar

Formula

m(n,m,d)=If[ n == m, (-1)^n, If[n == m - 1 || n == m + 1, -1, 0]]

Example

Triangular sequence:
{-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, 1},
{21, 21, -25, -25, 9, 9, -1, -1},
{34, 0, -65, 0, 42, 0, -11, 0, 1},
{-55, -55, 90, 90, -51, -51, 12, 12, -1, -1},
{-89, 0, 210, 0, -183, 0, 74, 0, -14, 0, 1}

Mathematica

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]

Crossrefs

Sequence in context: A099390 A297477 A370030 * A368514 A289229 A263097

Adjacent sequences: A124028 A124029 A124030 * A124032 A124033 A124034

Keyword

tabl,uned,sign

Author

Roger L. Bagula and Gary W. Adamson, Nov 01 2006

Status

approved