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A204164 Symmetric matrix based on f(i,j)=floor[(i+j)/2], by antidiagonals. 7

%I

%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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Last modified May 28 21:37 EDT 2020. Contains 334690 sequences. (Running on oeis4.)