A122771 Triangle read by rows, 0 <= k <= n: T(n,k) is the coefficient of x^k in the characteristic polynomial of I + A^(-1), where A is the n-step Fibonacci companion matrix and I is the identity matrix.

1, 2, -1, -1, -1, 1, 2, -2, 2, -1, -1, -2, 4, -3, 1, 2, -3, 6, -7, 4, -1, -1, -3, 9, -13, 11, -5, 1, 2, -4, 12, -22, 24, -16, 6, -1, -1, -4, 16, -34, 46, -40, 22, -7, 1, 2, -5, 20, -50, 80, -86, 62, -29, 8, -1, -1, -5, 25, -70, 130, -166, 148, -91, 37, -9, 1, 2, -6, 30, -95, 200, -296, 314, -239, 128, -46, 10, -1, -1, -6, 36, -125

Offset: 0

Author

Gary W. Adamson and Roger L. Bagula, Oct 20 2006

Keywords

• tabl
• sign

Comments

Here, the characteristic polynomial of a matrix M is defined as det(M-x*I).

The matrix I + A^(-1) for 2 <= n <= 6:

2 X 2: {{0, 1}, {1, 1}},

3 X 3: {{0, -1, 1}, {1, 1, 0}, {0, 1, 1}},

4 X 4: {{0, -1, -1, 1}, {1, 1, 0, 0}, {0, 1, 1, 0}, {0, 0, 1,1}},

5 X 5: {{0, -1, -1, -1, 1}, {1, 1, 0, 0, 0}, {0, 1, 1, 0, 0}, {0, 0, 1, 1, 0}, {0, 0, 0, 1,1}},

6 X 6: {{0, -1, -1, -1, -1, 1}, {1, 1, 0, 0, 0, 0}, {0, 1, 1, 0, 0, 0}, {0, 0, 1, 1, 0, 0}, {0, 0, 0, 1, 1, 0}, {0, 0, 0, 0, 1, 1}}

Examples

			Triangular array:
   1;
   2,  -1;
  -1,  -1,   1;
   2,  -2,   2,  -1;
  -1,  -2,   4,  -3,   1;
   2,  -3,   6,  -7,   4,  -1;
  -1,  -3,   9, -13,  11,  -5,   1;
   2,  -4,  12, -22,  24, -16,   6,  -1;
  -1,  -4,  16, -34,  46, -40,  22,  -7,   1;
   2,  -5,  20, -50,  80, -86,  62, -29,   8,  -1;
		

References

• Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002.
• Kappraff, J., Blackmore, D. and Adamson, G. "Phyllotaxis as a Dynamical System: A Study in Number." In Symmetry in Plants edited by R. V. Jean and D. Barabe. Singapore: World Scientific. (1996).

Links

• P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.

Programs

Mathematica

An[d_] := Table[If[n == d, 1, If[m == n + 1, 1, 0]], {n, 1, d}, {m, 1, d}];
Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[IdentityMatrix[d] + MatrixPower[An[d], -1], x], x], {d, 1, 20}]];
Flatten[%]

Python

from sympy import Matrix,eye
def A122771_row(n):
  if n==0: return [1]
  A=Matrix(n,n,lambda i,j:int(i==n-1 or i==j-1))
  p=(eye(n)+A.inv()).charpoly()
  return [(-1)**n*c for c in p.all_coeffs()[::-1]] # Pontus von Brömssen, May 01 2021

Extensions

Edited by Pontus von Brömssen, May 01 2021