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A204030
Symmetric matrix based on f(i,j) = gcd(i+1, j+1), by antidiagonals.
3
2, 1, 1, 2, 3, 2, 1, 1, 1, 1, 2, 1, 4, 1, 2, 1, 3, 1, 1, 3, 1, 2, 1, 2, 5, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 7, 2, 1, 2, 1, 2, 1, 3, 1, 5, 3, 1, 1, 3, 5, 1, 3, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1
OFFSET
1,1
COMMENTS
A204030 represents the matrix M given by f(i,j) = gcd(i+1, j+1) for i >= 1 and j >= 1. See A204031 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.
EXAMPLE
Northwest corner:
2 1 2 1 2 1 2 1
1 3 1 1 3 1 1 3
2 1 4 1 2 1 4 1
1 1 1 5 1 1 1 1
2 3 2 1 6 1 2 3
1 1 1 1 1 7 1 1
MATHEMATICA
f[i_, j_] := GCD[i + 1, j + 1];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8 X 8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 15}, {i, 1, n}]] (* A204030 *)
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[%] (* A204111 *)
TableForm[Table[c[n], {n, 1, 10}]]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jan 11 2012
STATUS
approved