_Clark Kimberling (ck6(AT)evansville.edu), _, Jan 11 2012

proposed

approved

editing

proposed

allocated for Clark KimberlingSymmetric matrix based on f(i,j)=max(ceiling(i/j),ceiling(j/i)), by antidiagonals.

1, 2, 2, 3, 1, 3, 4, 2, 2, 4, 5, 2, 1, 2, 5, 6, 3, 2, 2, 3, 6, 7, 3, 2, 1, 2, 3, 7, 8, 4, 2, 2, 2, 2, 4, 8, 9, 4, 3, 2, 1, 2, 3, 4, 9, 10, 5, 3, 2, 2, 2, 2, 3, 5, 10, 11, 5, 3, 2, 2, 1, 2, 2, 3, 5, 11, 12, 6, 4, 3, 2, 2, 2, 2, 3, 4, 6, 12, 13, 6, 4, 3, 2, 2, 1, 2, 2, 3, 4, 6, 13, 14, 7, 4, 3

1,2

A204143 represents the matrix M given by f(i,j)=max(ceiling(i/j),ceiling(j/i)) for i>=1 and j>=1. See A204144 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 3 4 5 6

2 1 2 2 3 3

3 2 1 2 2 2

4 2 2 1 2 2

5 3 2 2 1 2

6 3 2 2 2 1

f[i_, j_] := Max[Ceiling[i/j], Ceiling[j/i]];

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}]] (* A204143 *)

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[%] (* A204144 *)

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

Cf. A204144, A204016, A202453.

allocated

nonn,tabl

Clark Kimberling (ck6(AT)evansville.edu), Jan 11 2012

approved

editing

allocating

allocated

allocated for Clark Kimberling

allocating

approved