A204114 Symmetric matrix based on f(i,j) = gcd(L(i), L(j)), where L=A000032 (Lucas numbers), by antidiagonals.

1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 7, 1, 3, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 18, 1, 1, 4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 29, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1

Offset

1,5

Comments

A204114 represents the matrix M given by f(i,j) = gcd(L(i+1), L(j+1)) for i >= 1 and j >= 1. See A204115 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  3  1  1  1
  1  1  4  1  1
  1  1  1  7  1
  1  1  1  1 11

Mathematica

u[n_] := LucasL[n]
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}]]    (* A204114 *)
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[%]                   (* A204115 *)
TableForm[Table[c[n], {n, 1, 10}]]

Crossrefs

Cf. A204115, A204016, A202453.

Sequence in context: A333843 A339876 A204129 * A204131 A317941 A318667

Adjacent sequences: A204111 A204112 A204113 * A204115 A204116 A204117

Keyword

nonn,tabl

Author

Clark Kimberling, Jan 11 2012

Status

approved