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A204018
Symmetric matrix based on f(i,j)=1+max(j mod i, i mod j), by antidiagonals.
3
1, 2, 2, 2, 1, 2, 2, 3, 3, 2, 2, 3, 1, 3, 2, 2, 3, 4, 4, 3, 2, 2, 3, 4, 1, 4, 3, 2, 2, 3, 4, 5, 5, 4, 3, 2, 2, 3, 4, 5, 1, 5, 4, 3, 2, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 2, 3, 4, 5, 6, 1, 6, 5, 4, 3, 2, 2, 3, 4, 5, 6, 7, 7, 6, 5, 4, 3, 2, 2, 3, 4, 5, 6, 7, 1, 7, 6, 5, 4, 3, 2, 2, 3, 4, 5, 6, 7, 8, 8
OFFSET
1,2
COMMENTS
A204018 represents the matrix M given by f(i,j)=max{1+(j mod i), 1+( i mod j)} for i>=1 and j>=1. See A204019 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.
EXAMPLE
Northwest corner:
1 2 2 2 2 2
2 1 3 3 3 3
2 3 1 4 4 4
2 3 4 1 5 5
2 3 4 5 1 6
2 3 4 5 6 1
MATHEMATICA
f[i_, j_] := 1 + Max[Mod[i, j], Mod[j, i]];
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, 15}, {i, 1, n}]] (* A204018 *)
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[%] (* A204019 *)
TableForm[Table[c[n], {n, 1, 10}]]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jan 11 2012
STATUS
approved