A204116 Symmetric matrix based on f(i,j) = gcd(2^i-1, 2^j-1), by antidiagonals.

1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 7, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 15, 1, 3, 1, 1, 1, 7, 1, 1, 7, 1, 1, 1, 3, 1, 3, 31, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 7, 15, 1, 63, 1, 15, 7, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 127, 3, 1, 3, 1, 3, 1, 1, 1, 7, 1

Offset: 1

Author

Clark Kimberling, Jan 11 2012

Keywords

• nonn
• tabl

Comments

A204116 represents the matrix M given by f(i,j) = gcd(2^i-1, 2^j-1) for i >= 1 and j >= 1. See A204117 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.

Examples

			Northwest corner:
  1  1  1  1
  1  3  1  3
  1  1  7  1
  1  3  1 15
		

Crossrefs

Cf. A204117, A204016, A202453.

Programs

Mathematica

f[i_, j_] := GCD[2^i - 1, 2^j - 1];
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}]]  (* A204116 *)
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[%]                 (* A204117 *)
TableForm[Table[c[n], {n, 1, 10}]]