A368734 Four-column table read by rows where row n lists the entries of the 2 X 2 matrix M(n) used to form Bird tree and Drib tree rationals.

1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 2, 0, 1, 1, 0, 2, 1, 2, 1, 1, 1, 1, 2, 2, 3, 2, 3, 1, 1, 0, 1, 1, 3, 1, 3, 1, 2, 2, 1, 3, 1, 3, 1, 1, 0, 1, 1, 3, 2, 3, 2, 2, 1, 2, 3, 3, 5, 3, 5, 1, 2, 1, 1, 3, 4, 3, 4, 2, 3, 1, 3, 1, 4, 1, 4, 0, 1, 1, 2, 2, 5

Offset: 1

Author

Philippe Deléham, Jan 04 2024

Keywords

• nonn
• easy
• tabf

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;

...

Examples

			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); ...}.
M(38) = [2, 3; 5, 7] ; Det(M(38)) = 2*7-3*5 = -1; 2+5 = 7 = A162909(38); 3+7 = 10 = A162910(38); 2+3 = 5 = A162911(38); 5+7 = 12 = A162912(38).
		

Crossrefs

Cf. A000045, A054429, A122872, A162909, A162910, A162911, A162912.

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(A122872(n,k)).

M(2^n) = [F(n-1), F(n); F(n), F(n+1)], F(n) = Fibonacci(n) = A000045(n).