OFFSET
1,16
COMMENTS
Row n is x(n), y(n), z(n), t(n) and the matrix is M(n) = [x(n), y(n) ; z(n), t(n)].
Bird tree rationals are formed by multiplication on the right of a row vector [1,1]*M(n) = [A162909(n), A162910(n)].
Drib tree rationals are formed by multiplication on the left of a column vector M(n)*[1;1] = [A162911(n), A162912(n)].
The two matrix rows f(n) = (x(n), y(n)) and g(n) = (z(n), t(n)) start f(1) = (1,0) and g(1) = (0,1) and then satisfy f(2*n) = g(n); g(2*n) = f(2*n+1) = f(n) + g(n); g(2*n+1) = f(n).
If the terms x(n), y(n), z(n) or t(n) are written as an array (left-aligned fashion) with rows of length 2^m, m = 0; 1, 2, 3, ... then each column k is a Fibonacci-type sequence.
For x(n):
1;
0, 1;
1, 1, 1, 2;
1, 2, 0, 1, 2, 3, 1, 3;
2, 3, 1, 3, 1, 1, 1, 2, 3, 5, 1, 4, 3, 4, 2, 5;
...
For y(n):
0;
1, 1;
1, 2, 0, 1;
2, 3, 1, 3, 1, 1, 1, 2;
3, 5, 1, 4, 3, 4, 2, 5, 1, 2, 0, 1, 2, 3, 1, 3;
...
For z(n):
0;
1, 1;
1, 0, 2, 1;
2, 1, 1, 1, 3, 1, 3, 2;
3, 1, 3, 2, 1, 0, 2, 1, 5, 2, 4, 3, 4, 1, 5, 3;
...
For t(n):
1;
1, 0;
2, 1, 1, 1;
3, 1, 3, 2, 1, 0, 2, 1;
5, 2, 4, 3, 4, 1, 5, 3, 2, 1, 1, 1, 3, 1, 3, 2;
...
FORMULA
x(n) + z(n) = A162909(n).
y(n) + t(n) = A162910(n).
x(n) + y(n) = A162911(n).
z(n) + t(n) = A162912(n).
Det(M(n)) = (-1)^p if 2^p <= n < 2^(p+1).
x(A054429(n)) = t(n).
y(A054429(n)) = z(n).
z(A054429(n)) = y(n).
t(A054429(n)) = x(n).
M(k)*M(n) = M(A(122872(n,k)).
M(2^n) = [F(n-1), F(n); F(n), F(n+1)], F(n) = Fibonacci(n) = A000045(n).
EXAMPLE
The table begins:
| f(n) | g(n)
n | x(n)| y(n)| z(n)| t(n)
1 | 1 | 0 | 0 | 1
2 | 0 | 1 | 1 | 1
3 | 1 | 1 | 1 | 0
4 | 1 | 1 | 1 | 2
5 | 1 | 2 | 0 | 1
6 | 1 | 0 | 2 | 1
7 | 2 | 1 | 1 | 1
.
For n >= 1, f(n) = {(1,0); (0,1); (1,1); (1,1); (1,2); (1,0); ...}.
For n >= 1, g(n) = {(0,1); (1,1); (1,0); (1,2); (0,1); (2,1); ...}.
CROSSREFS
KEYWORD
nonn,easy,tabf
AUTHOR
Philippe Deléham, Jan 04 2024
STATUS
approved