|
|
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.
|
|
2
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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
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()
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|