|
|
A123973
|
|
Sequence of tridiagonal matrices with one center zero terminal that give a triangular sequence from the characteristic polynomials based on the 3 X 3 matrix type: {{1, -1, 0}, {-1, 1, -1}, {0, -1, 0}}.
|
|
0
|
|
|
0, 0, -1, -1, -1, 1, -1, 1, 2, -1, 0, 3, 0, -3, 1, 1, 2, -5, -2, 4, -1, 1, -2, -7, 6, 5, -5, 1, 0, -5, 0, 15, -5, -9, 6, -1, -1, -3, 12, 9, -25, 1, 14, -7, 1, -1, 3, 15, -18, -29, 35, 7, -20, 8, -1, 0, 7, 0, -42, 14, 63, -42, -20, 27, -9, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,9
|
|
COMMENTS
|
Matrices: {{0}}, {{1, -1}, {-1, 0}}, {{1, -1, 0}, {-1, 1, -1}, {0, -1, 0}}, {{1, -1, 0, 0}, {-1, 1, -1, 0}, {0, -1, 1, -1}, {0, 0, -1, 0}}, {{1, -1, 0, 0, 0}, {-1, 1, -1, 0, 0}, {0, -1, 1, -1, 0}, {0, 0, -1, 1, -1}, {0, 0, 0, -1, 0}}, {{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, 0}} Determinants ( not all Sl(3,Z) and invertable): Table[Det[M[d]], {d, 1, 10}] {0, -1, -1, 0, 1, 1, 0, -1, -1, 0}
|
|
LINKS
|
|
|
FORMULA
|
t(n,m,d)=If[ n == m && n < d && m < d, 1, If[n == m - 1 || n == m + 1, -1, If[n == m == d, 0, 0]]]
|
|
EXAMPLE
|
Triangle begins:
{0},
{0, -1},
{-1, -1, 1},
{-1, 1, 2, -1},
{0, 3, 0, -3, 1},
{1, 2, -5, -2, 4, -1},
{1, -2, -7, 6, 5, -5, 1},
{0, -5, 0, 15, -5, -9, 6, -1},
{-1, -3, 12, 9, -25, 1, 14, -7, 1},
{-1, 3, 15, -18, -29, 35,7, -20, 8, -1},
{0, 7, 0, -42, 14, 63, -42, -20, 27, -9, 1}
Some of the polynomials are Steinbach.
|
|
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, 0, 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
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
Looking at the triangle suggests that the very first term should be 1, not 0. - N. J. A. Sloane, Nov 01 2006
|
|
STATUS
|
approved
|
|
|
|