A122185 Cube of the Steinbach matrix as a characteristic polynomial triangular array.

1, 1, -1, -1, -4, 1, -1, 4, 11, -1, 1, 4, -21, -23, 1, 1, -4, -31, 60, 42, -1, -1, -4, 41, 100, -171, -69, 1, -1, 4, 51, -140, -400, 381, 106, -1, 1, 4, -61, -180, 729, 1060, -823, -154, 1, 1, -4, -71, 220, 1158, -2136, -3032, 1561, 215, -1, -1, -4, 81, 260, -1687, -3612, 7721, 6887, -2874, -290, 1, -1, 4, 91, -300, -2316

Offset

1,5

Comments

After finding the relationship of A[i,j]^2=Min[i,j] (A122073) and I[n]-A[i,j] (A122160), A[i,j]^3 suggested itself. Matrices: {{3, 2}, {2, 1}}, {{6, 5, 3}, {5, 4,2}, {3, 2, 1}}, {{10, 9, 7, 4}, {9, 8, 6, 3}, {7, 6, 4, 2}, {4, 3, 2,1}}, {{15, 14, 12, 9, 5}, {14, 13, 11, 8, 4}, {12, 11, 9,6, 3}, {9, 8, 6, 4, 2}, {5, 4, 3, 2, 1}}

Links

Table of n, a(n) for n=1..71.

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.

Formula

A(i,j)^3-->P(n,k) P(n,k)->T(n,m)

Example

{1},
{1, -1},
{-1, -4, 1},
{-1, 4, 11, -1},
{1, 4, -21, -23, 1},
{1, -4, -31, 60, 42, -1},
{-1, -4, 41, 100, -171, -69, 1},
{-1, 4, 51, -140, -400, 381, 106, -1}

Mathematica

An[d_] := Table[If[n + m - 1 > d, 0, 1], {n, 1, d}, {m, 1, d}]; Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[MatrixPower[An[d], 3], x], x], {d, 1, 20}]]; Flatten[%]

Crossrefs

Sequence in context: A001638 A133826 A209565 * A350000 A193752 A136680

Adjacent sequences: A122182 A122183 A122184 * A122186 A122187 A122188

Keyword

uned,tabl,sign

Author

Gary W. Adamson and Roger L. Bagula, Oct 17 2006

Status

approved