%I #5 Mar 30 2012 18:58:07

%S 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,

%T 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,

%U 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

%N Symmetric matrix based on f(i,j)=ceiling[(i+j)/2], by antidiagonals.

%C 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.

%e Northwest corner:

%e 1 2 2 3 3 4 4 5

%e 2 2 3 3 4 4 5 5

%e 2 3 3 4 4 5 5 6

%e 3 3 4 4 5 5 6 6

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

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

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

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

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

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

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

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

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

%t Flatten[%] (* A204167 *)

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

%Y Cf. A204167, A204016, A202453.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Jan 12 2012