A124035 Tridiagonal matrices of central ones with lower negative one to give a triangular sequence: first element is negative one. m(n,m,d)=If[ n == m && n < d && m < d, 1, If[n == m - 1 || n == m + 1, -1, If[n == m == d, -1, 0]].

-1, -1, -1, -2, 0, 1, -1, 3, 1, -1, 1, 4, -3, -2, 1, 2, 0, -8, 2, 3, -1, 1, -6, -5, 12, 0, -4, 1, -1, -7, 9, 15, -15, -3, 5, -1, -2, 0, 21, -6, -30, 16, 7, -6, 1, -1, 9, 12, -42, -9, 49, -14, -12, 7, -1, 1, 10, -18, -48, 63, 42, -70, 8, 18, -8, 1

Offset

1,4

Comments

The 4 X 4 is a g(u,v) type of matrix where Minkowski is n(u,v)={1,1,1,-1}: single hyperbolic index. 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}

Links

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

Formula

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

Example

Triangular sequence:
{{-1}},
{-1, -1},
{-2, 0, 1},
{-1, 3, 1, -1},
{1, 4, -3, -2,1},
{2, 0, -8, 2, 3, -1},
{1, -6, -5, 12, 0, -4, 1},
{-1, -7, 9, 15, -15, -3, 5, -1},
{-2, 0, 21, -6, -30, 16, 7, -6, 1},

Mathematica

T[n_, m_, d_] := If[ n == m && n < d && m < d, 1, If[n == m - 1 || n == m + 1, -1, If[n == m == d, -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: A321258 A331510 A319854 * A204184 A157897 A213910

Adjacent sequences: A124032 A124033 A124034 * A124036 A124037 A124038

Keyword

uned,sign,tabl

Author

Roger L. Bagula, Nov 02 2006

Status

approved