A122073 Triangular array from Steinbach matrices plus their squares as characteristic polynomials: M[i,j]=A[i,j]+A[i,j]^2: A[i,j]^2=Min[i,j]; CharacteristicPolynomial[M[i,j]];.

1, 2, -1, 0, -4, 1, 2, -9, 8, -1, -2, -3, 19, -12, 1, -4, -6, 47, -55, 18, -1, 2, 15, 0, -88, 93, -24, 1, 2, 23, -7, -190, 324, -182, 32, -1, 0, -12, -63, 62, 332, -554, 274, -40, 1, 2, -9, -108, 133, 678, -1642, 1346, -450, 50, -1, -2, -11, 55, 276, -463, -1129, 2832, -2128, 630, -60, 1, -4, -30, 71, 543, -1044, -2204, 7761

Offset: 1

Author

Gary W. Adamson and Roger L. Bagula, Oct 16 2006

Keywords

• tabl
• uned
• sign

Comments

Based on the idea that the Steinbach matrices form a "golden Field". Matrices are: {{2, 2}, {2, 2}}, {{2, 2, 2}, {2, 3, 2}, {2, 2, 3}}, {{2, 2, 2, 2}, {2, 3, 3, 2}, {2, 3, 3, 3}, {2, 2, 3, 4}}, {{2, 2, 2, 2, 2}, {2, 3, 3, 3, 2}, {2, 3, 4, 3, 3}, {2, 3, 3, 4, 4}, {2, 2, 3, 4, 5}}, {{2, 2, 2, 2, 2, 2}, {2,3, 3, 3, 3, 2}, {2, 3, 4, 4, 3, 3}, {2, 3, 4, 4, 4, 4}, {2, 3, 3, 4, 5, 5}, {2, 2, 3, 4, 5, 6}}

Examples

			{1},
{2, -1},
{0, -4, 1},
{2, -9, 8, -1},
{-2, -3, 19, -12, 1},
{-4, -6,47, -55, 18, -1}
{2, 15, 0, -88, 93, -24, 1},
{2, 23, -7, -190, 324, -182, 32, -1},
{0, -12, -63, 62, 332, -554, 274, -40, 1}
		

Links

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

Crossrefs

Cf. A038223, A038223, A076756, A054142.

Programs

Mathematica

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

Formula

d-th level M(i,j)->An[d]; T(n,m)=CoefficientList[CharacteristicPolynomial[An[d], x], x]