A204127 Symmetric matrix based on f(i,j)=(F(i+1) if i=j and 1 otherwise), where F=A000045 (Fibonacci numbers), by antidiagonals.

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.

Links

Table of n, a(n) for n=1..98.

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

Cf. A204128, A204016, A202453.

Sequence in context: A261795 A306345 A204125 * A225174 A059895 A368328

Adjacent sequences: A204124 A204125 A204126 * A204128 A204129 A204130

Keyword

nonn,tabl

Author

Clark Kimberling, Jan 11 2012

Status

approved