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A204127
Symmetric matrix based on f(i,j)=(F(i+1) if i=j and 1 otherwise), where F=A000045 (Fibonacci numbers), by antidiagonals.
3
1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 21, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
OFFSET
1,5
COMMENTS
A204127 represents the matrix M given by f(i,j)=(F(i+1) if i=j and 1 otherwise) for i>=1 and j>=1. See A204128 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 1 1 1 1
1 1 3 1 1 1
1 1 1 5 1 1
1 1 1 1 8 1
1 1 1 1 1 13
MATHEMATICA
f[i_, j_] := 1; f[i_, i_] := Fibonacci[i + 1];
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}]] (* A204127 *)
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[%] (* A204128 *)
TableForm[Table[c[n], {n, 1, 10}]]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jan 11 2012
STATUS
approved