login
A204026
Symmetric matrix based on f(i,j)=min(F(i+1),F(j+1)), where F=A000045 (Fibonacci numbers), by antidiagonals.
4
1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 3, 2, 1, 1, 2, 3, 5, 3, 2, 1, 1, 2, 3, 5, 5, 3, 2, 1, 1, 2, 3, 5, 8, 5, 3, 2, 1, 1, 2, 3, 5, 8, 8, 5, 3, 2, 1, 1, 2, 3, 5, 8, 13, 8, 5, 3, 2, 1, 1, 2, 3, 5, 8, 13, 13, 8, 5, 3, 2, 1, 1, 2, 3, 5, 8, 13, 21, 13, 8, 5, 3, 2, 1, 1, 2, 3, 5, 8
OFFSET
1,5
COMMENTS
A204026 represents the matrix M given by f(i,j)=min(F(i+1),F(j+1)) for i>=1 and j>=1. See A204027 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 1 1 1 1 1
1 2 2 2 2 2
1 2 3 3 3 3
1 2 3 5 5 5
1 2 3 5 8 8
1 2 3 5 8 13
MATHEMATICA
f[i_, j_] := Min[Fibonacci[i + 1], Fibonacci[j + 1]]
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}]] (* A204026 *)
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[%] (* A204027 *)
TableForm[Table[c[n], {n, 1, 10}]]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jan 11 2012
STATUS
approved