A124036 Alternating ones and twos tridiagonal matrices ( columns of 1's and twos) to give a triangular sequence: m(n,m,d)=If[ n == m, 1 + (1 - (-1)^(n + 1))/2, If[n == m - 1 || n == m + 1, 1 + (1 - (-1)^n)/2, 0]].

1, 1, -1, 0, -3, 1, -2, -1, 4, -1, -4, 6, 7, -6, 1, 0, 12, -7, -11, 7, -1, 8, 12, -40, -3, 23, -9, 1, 8, -20, -38, 59, 12, -30, 10, -1, 0, -72, 24, 162, -81, -54, 48, -12, 1, -16, -32, 172, 20, -267, 87, 82, -58, 13, -1, -32, 96, 328, -456, -392, 549, -19, -174, 82, -15, 1

Offset

1,5

Comments

Matrices: {{1}}, {{1, 2}, {1, 2}}, {{1, 2,0}, {1, 2, 1}, {0, 2, 1}}, {{1, 2, 0, 0}, {1, 2, 1, 0}, {0, 2, 1, 2}, {0, 0, 1, 2}}, {{1, 2, 0, 0, 0}, {1, 2, 1, 0,0}, {0, 2, 1, 2, 0}, {0, 0,1, 2, 1}, {0, 0, 0, 2, 1}}, {{1, 2, 0, 0, 0, 0}, {1, 2, 1, 0, 0, 0}, {0, 2,1, 2, 0, 0}, {0, 0, 1, 2, 1, 0}, {0, 0, 0, 2, 1, 2}, {0, 0, 0, 0, 1, 2}} Large roots: Table[x /. NSolve[Det[M[d] - x*IdentityMatrix[d]] == 0, x][[d]], {d, 1, 10}] {1., 3., 3.56155, 3.84224, 4., 4.09691, 4.16053, 4.20447, 4.23607, 4.25953}

Links

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

Formula

m(n,m,d)=If[ n == m, 1 + (1 - (-1)^(n + 1))/2, If[n == m - 1 || n == m + 1, 1 + (1 - (-1)^n)/2, 0]]

Example

Triangular sequence:
{{1}},
{1, -1},
{0, -3, 1},
{-2, -1, 4, -1},
{-4, 6, 7, -6, 1},
{0, 12, -7, -11, 7, -1},
{8, 12, -40, -3, 23, -9, 1},
{8, -20, -38, 59, 12, -30, 10, -1},
{0, -72, 24, 162, -81, -54, 48, -12, 1},
{-16, -32, 172, 20, -267,87, 82, -58, 13, -1},
{-32, 96, 328, -456, -392, 549, -19, -174, 82, -15, 1}

Mathematica

T[n_, m_, d_] := If[ n == m, 1 + ( 1 - (-1)^(n + 1))/2, If[n == m - 1 || n == m + 1, 1 + (1 - (-1)^n)/2, 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: A166866 A370220 A177343 * A182481 A066743 A257915

Adjacent sequences: A124033 A124034 A124035 * A124037 A124038 A124039

Keyword

uned,sign

Author

Gary W. Adamson and Roger L. Bagula, Nov 02 2006

Status

approved