_Clark Kimberling (ck6(AT)evansville.edu), _, Jan 12 2012

proposed

approved

editing

proposed

allocated for Clark KimberlingSymmetric matrix based on f(i,j)=ceiling[(i+j)/2], by antidiagonals.

1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8

1,2

A204166 represents the matrix M given by f(i,j)=ceiling[(i+j)/2] for i>=1 and j>=1. See A204167 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.

Northwest corner:

1 2 2 3 3 4 4 5

2 2 3 3 4 4 5 5

2 3 3 4 4 5 5 6

3 3 4 4 5 5 6 6

f[i_, j_] := Ceiling[(i + j)/2];

m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]

TableForm[m[8]] (* 8x8 principal submatrix *)

Flatten[Table[f[i, n + 1 - i],

{n, 1, 15}, {i, 1, n}]] (* A204166 *)

p[n_] := CharacteristicPolynomial[m[n], x];

c[n_] := CoefficientList[p[n], x]

TableForm[Flatten[Table[p[n], {n, 1, 10}]]]

Table[c[n], {n, 1, 12}]

Flatten[%] (* A204167 *)

TableForm[Table[c[n], {n, 1, 10}]]

Cf. A204167, A204016, A202453.

allocated

nonn,tabl

Clark Kimberling (ck6(AT)evansville.edu), Jan 12 2012

approved

editing

allocated for Clark Kimberling

allocated

approved