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%I #6 Mar 30 2012 18:58:07
%S 1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,
%T 4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,
%U 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
%N Symmetric matrix based on f(i,j)=floor[(i+j)/2], by antidiagonals.
%C A204164 represents the matrix M given by f(i,j)=floor[(i+j)/2] for i>=1 and j>=1. See A204165 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 1 2 2 3 3 4 4
%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_] := Floor[(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}]] (* A204164 *)
%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[%] (* A204165 *)
%t TableForm[Table[c[n], {n, 1, 10}]]
%Y Cf. A204165, A204016, A202453.
%K nonn,tabl
%O 1,4
%A _Clark Kimberling_, Jan 12 2012
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