A131309 - OEIS

%I #15 Mar 01 2022 07:28:50

%S 1,1,1,0,1,1,0,1,1,0,1,0,1,1,0,1,1,0,1,0,1,1,0,1,1,0,1,0,1,1,0,1,0

%N Rabbit-like sequence for phi^2.

%C The terms given can be computed as the iterates of the morphism 1 -> 110, 0 -> 10, with axiom 1, concatenated. - _Joerg Arndt_, Mar 01 2022

%C Ratio of 1's to 0's tends to phi^2, by way of example, in the subset of 8 terms (1, 1, 0, 1, 1, 0, 1, 0), there are five 1's and three 0's.

%C Subsets have A001906: (1, 3, 8, 21, ...) terms, being partial sums of A027941: (1, 4, 12, 33, ...). After 33 total terms, there are (1 + 3 + 8) zeros and (1 + 2 + 5 + 13) = 21 ones; with the ratio of ones to zeros tending to phi^2 = 2.618...

%F Substitution rules T => 2T = t; t => T + t; are derived directly from the matrix generator [2,1; 1,0] (eigenvalue phi^2). Then substitute 1 for T and 0 for t.

%e By rows, we get:

%e   1;

%e   1, 1, 0;

%e   1, 1, 0, 1, 1, 0, 1, 0;

%e   ...

%e Then append n-th row to the end of (n-1)-th row, forming a continuous string.

%Y Cf. A027941, A001906, A104457 (phi^2).

%K nonn,tabf,more,uned

%O 0,1

%A _Gary W. Adamson_, Jun 27 2007