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

1,16

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;

...

Links

Table of n, a(n) for n=1..88.

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); ...}.
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.

Sequence in context: A358360 A094114 A104607 * A120728 A092149 A303975

Adjacent sequences: A368731 A368732 A368733 * A368735 A368736 A368737

Keyword

nonn,easy,tabf

Author

Philippe Deléham, Jan 04 2024

Status

approved