A124020 New tetradiagonal form matrix as triangular sequence from solution of : X(n,m)=Steinbach(n,m)^(-1).tri-Antidiagonal_1(n,n).

1, 1, -1, 0, -3, 1, -1, -4, 5, -1, -1, -2, 12, -7, 1, 0, 2, 17, -24, 9, -1, 1, 4, 13, -52, 40, -11, 1, 1, 1, 0, -76, 115, -60, 13, -1, 0, -5, -10, -72, 235, -214, 84, -15, 1, -1, -8, -2, -34, 352, -554, 357, -112, 17, -1, -1, -4, 24, 2, 383, -1092, 1113, -552, 144, -19, 1, 0, 4, 46, -24, 295, -1673, 2688, -2008, 807, -180, 21, -1

Offset

1,5

Comments

Matrices {{1}}, {{1, -1}, {-2, 2}}, {{1, -1, 0}, {-2, 2, -1}, {1, -2, 2}}, {{1, -1, 0, 0}, {-2, 2, -1, 0}, {1, -2, 2, -1}, {0, 1, -2, 2}}, {{1, -1, 0, 0, 0}, {-2, 2, -1, 0, 0}, {1, -2, 2, -1, 0}, {0, 1, -2, 2, -1}, {0, 0, 1, -2, 2}} Large root sequence: a0 = Table[x /. NSolve[CharacteristicPolynomial[Xn[d], x] == 0, x][[d]], {d, 1, 20}] {1., 3., 3.91223, 4.37167, 4.62826, 4.78478, 4.88683, 4.95691, 5.00703, 5.0441, 5.07226, 5.09415, 5.1115, 5.12547, 5.1369, 5.14636, 5.15428, 5.16097, 5.16668, 5.17159} Determinant sequence is: Table[Det[Xn[d]], {d, 1, 20}] {1,0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0}

Links

Table of n, a(n) for n=1..78.

Formula

Steinbach(n,m)=If[n + m - 1 > d, 0, 1]; tri-Antidiagonal_1(n,m)=If[n + m - 1 == d, 1, If[n + m == d, -1, If[n + m - 2 == d, -1, 0]]]; X(n,m)=Steinbach(n,m)^(-1).tri-Antidiagonal2(n,n)

Example

triangular sequence:
{1},
{1, -1},
{0, -3, 1},
{-1, -4,5, -1},
{-1, -2, 12, -7, 1},
{0, 2, 17, -24, 9, -1},
{1, 4,13, -52, 40, -11,1},
{1, 1, 0, -76, 115, -60, 13, -1},
{0, -5, -10, -72, 235, -214, 84, -15,1}

Mathematica

An[d_] := Table[If[n + m - 1 > d, 0, 1], {n, 1, d}, {m, 1, d}]; Bn[d_] := Table[If[n + m - 1 == d, 1, If[n + m == d, -1, If[n + m - 2 == d, -1, 0]]], {n, 1, d}, {m, 1, d}]; Xn[d_] := MatrixPower[An[d], -1].Bn[d]; a = Join[Xn[1], Table[CoefficientList[CharacteristicPolynomial[Xn[d], x], x], {d, 1, 20}]]; Flatten[%]

Crossrefs

Sequence in context: A262191 A334844 A296457 * A124234 A135226 A104730

Adjacent sequences: A124017 A124018 A124019 * A124021 A124022 A124023

Keyword

sign,tabl,uned

Author

Roger L. Bagula and Gary W. Adamson, Oct 31 2006

Status

approved