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%I #37 Jan 12 2024 22:42:09
%S 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,
%T 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,
%U 1,1,3,4,3,4,2,3,1,3,1,4,1,4,0,1,1,2,2,5
%N 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.
%C 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)].
%C Bird tree rationals are formed by multiplication on the right of a row vector [1,1]*M(n) = [A162909(n), A162910(n)].
%C Drib tree rationals are formed by multiplication on the left of a column vector M(n)*[1;1] = [A162911(n), A162912(n)].
%C 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).
%C 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.
%C For x(n):
%C 1;
%C 0, 1;
%C 1, 1, 1, 2;
%C 1, 2, 0, 1, 2, 3, 1, 3;
%C 2, 3, 1, 3, 1, 1, 1, 2, 3, 5, 1, 4, 3, 4, 2, 5;
%C ...
%C For y(n):
%C 0;
%C 1, 1;
%C 1, 2, 0, 1;
%C 2, 3, 1, 3, 1, 1, 1, 2;
%C 3, 5, 1, 4, 3, 4, 2, 5, 1, 2, 0, 1, 2, 3, 1, 3;
%C ...
%C For z(n):
%C 0;
%C 1, 1;
%C 1, 0, 2, 1;
%C 2, 1, 1, 1, 3, 1, 3, 2;
%C 3, 1, 3, 2, 1, 0, 2, 1, 5, 2, 4, 3, 4, 1, 5, 3;
%C ...
%C For t(n):
%C 1;
%C 1, 0;
%C 2, 1, 1, 1;
%C 3, 1, 3, 2, 1, 0, 2, 1;
%C 5, 2, 4, 3, 4, 1, 5, 3, 2, 1, 1, 1, 3, 1, 3, 2;
%C ...
%F x(n) + z(n) = A162909(n).
%F y(n) + t(n) = A162910(n).
%F x(n) + y(n) = A162911(n).
%F z(n) + t(n) = A162912(n).
%F Det(M(n)) = (-1)^p if 2^p <= n < 2^(p+1).
%F x(A054429(n)) = t(n).
%F y(A054429(n)) = z(n).
%F z(A054429(n)) = y(n).
%F t(A054429(n)) = x(n).
%F M(k)*M(n) = M(A(122872(n,k)).
%F M(2^n) = [F(n-1), F(n); F(n), F(n+1)], F(n) = Fibonacci(n) = A000045(n).
%e The table begins:
%e | f(n) | g(n)
%e n | x(n)| y(n)| z(n)| t(n)
%e 1 | 1 | 0 | 0 | 1
%e 2 | 0 | 1 | 1 | 1
%e 3 | 1 | 1 | 1 | 0
%e 4 | 1 | 1 | 1 | 2
%e 5 | 1 | 2 | 0 | 1
%e 6 | 1 | 0 | 2 | 1
%e 7 | 2 | 1 | 1 | 1
%e .
%e For n >= 1, f(n) = {(1,0); (0,1); (1,1); (1,1); (1,2); (1,0); ...}.
%e For n >= 1, g(n) = {(0,1); (1,1); (1,0); (1,2); (0,1); (2,1); ...}.
%e 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).
%Y Cf. A000045, A054429, A122872, A162909, A162910, A162911, A162912.
%K nonn,easy,tabf
%O 1,16
%A _Philippe Deléham_, Jan 04 2024
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