A203998 Symmetric matrix based on f(i,j)=max{i(j+1)-1,j(i+1)-1}, by antidiagonals.

1, 3, 3, 5, 5, 5, 7, 8, 8, 7, 9, 11, 11, 11, 9, 11, 14, 15, 15, 14, 11, 13, 17, 19, 19, 19, 17, 13, 15, 20, 23, 24, 24, 23, 20, 15, 17, 23, 27, 29, 29, 29, 27, 23, 17, 19, 26, 31, 34, 35, 35, 34, 31, 26, 19, 21, 29, 35, 39, 41, 41, 41, 39, 35, 29, 21, 23, 32, 39, 44

Offset: 1

Author

Clark Kimberling, Jan 09 2012

Keywords

• nonn
• tabl

Comments

A203998 represents the matrix M given by f(i,j)=max{i(j+1)-1,j(i+1)-1}for i>=1 and j>=1. See A203999 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

Examples

			Northwest corner:
1...3....5....7....9
3...5....8....11...14
5...8....11...15...19
7...11...15...19...24
		

Crossrefs

Cf. A203999, A202453.

Programs

Mathematica

f[i_, j_] := Max[i (j + 1) - 1, j (i + 1) - 1];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[6]] (* 6x6 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 12}, {i, 1, n}]]    (* A203998 *)
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[%]   (* A203999 *)
TableForm[Table[c[n], {n, 1, 10}]]