A204113 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the matrix at A204112, given by f(i,j) = gcd(F(i+1), F(j+1)), where F=A000045 (Fibonacci numbers).

1, -1, 1, -3, 1, 2, -8, 6, -1, 8, -36, 35, -11, 1, 48, -232, 274, -116, 19, -1, 576, -2880, 3620, -1728, 358, -32, 1, 10368, -52992, 70632, -37192, 8906, -1016, 53, -1, 331776, -1716480, 2354112, -1294352, 332812, -42924, 2805

Offset: 1

Author

Clark Kimberling, Jan 11 2012

Keywords

• tabl
• sign

Comments

Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix. The zeros of p(n) are real, and they interlace the zeros of p(n+1). See A202605 and A204016 for guides to related sequences.

Examples

			Top of the array:
  1,  -1;
  1,  -3,   1;
  2,  -8,   6,  -1;
  8, -36,  35, -11,   1;
		

References

• (For references regarding interlacing roots, see A202605.)

Crossrefs

Cf. A204112, A202605, A204016.

Programs

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}]]