A204112 Symmetric matrix based on f(i,j) = gcd(F(i+1), F(j+1)), where F=A000045 (Fibonacci numbers), by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 8, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 2, 1, 5, 2, 1, 1, 2, 5, 1, 2, 1, 1, 1, 3, 1, 1, 1, 21, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset

1,5

Comments

A204112 represents the matrix M given by f(i,j) = gcd(F(i+1), F(j+1)) for i >= 1 and j >= 1. See A204113 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  2  1
  1  1  3  1  1  1
  1  1  1  5  1  1
  1  2  1  1  8  1
  1  1  1  1  1 13

Mathematica

u[n_] := Fibonacci[n + 1]
f[i_, j_] := GCD[u[i], u[j]];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8 X 8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 15}, {i, 1, n}]]    (* A204112 *)
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[%]                   (* A204113 *)
TableForm[Table[c[n], {n, 1, 10}]]

Crossrefs

Cf. A204113, A204016, A202453.

Sequence in context: A321125 A205617 A359888 * A186027 A359250 A322482

Adjacent sequences: A204109 A204110 A204111 * A204113 A204114 A204115

Keyword

nonn,tabl

Author

Clark Kimberling, Jan 11 2012

Status

approved