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A204123
Symmetric matrix based on f(i,j)=max([i/j],[j/i]), where [ ]=floor, by antidiagonals.
4
1, 2, 2, 3, 1, 3, 4, 1, 1, 4, 5, 2, 1, 2, 5, 6, 2, 1, 1, 2, 6, 7, 3, 1, 1, 1, 3, 7, 8, 3, 2, 1, 1, 2, 3, 8, 9, 4, 2, 1, 1, 1, 2, 4, 9, 10, 4, 2, 1, 1, 1, 1, 2, 4, 10, 11, 5, 3, 2, 1, 1, 1, 2, 3, 5, 11, 12, 5, 3, 2, 1, 1, 1, 1, 2, 3, 5, 12, 13, 6, 3, 2, 1, 1, 1, 1, 1, 2, 3, 6, 13, 14, 6, 4, 2
OFFSET
1,2
COMMENTS
This sequence represents the matrix M given by f(i,j)=max([i/j],[j/i]) for i>=1 and j>=1. See A204124 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 3 4 5 6
2 1 1 2 2 3
3 1 1 1 1 2
4 2 1 1 1 1
5 2 1 1 1 1
6 3 2 1 1 1
MATHEMATICA
f[i_, j_] := Max[Floor[i/j], Floor[j/i]];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 15}, {i, 1, n}]] (* A204123 *)
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[%] (* A204124 *)
TableForm[Table[c[n], {n, 1, 10}]]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jan 11 2012
STATUS
approved