OFFSET
0,2
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}}
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.
EXAMPLE
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;
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[%]
PROG
(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
CROSSREFS
KEYWORD
tabl,sign
AUTHOR
Gary W. Adamson and Roger L. Bagula, Oct 20 2006
EXTENSIONS
Edited by Pontus von Brömssen, May 01 2021
STATUS
approved