A131309 Rabbit-like sequence for phi^2.

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

Offset

0,1

Comments

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

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.

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...

Links

Table of n, a(n) for n=0..32.

Formula

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.

Example

By rows, we get:
  1;
  1, 1, 0;
  1, 1, 0, 1, 1, 0, 1, 0;
  ...
Then append n-th row to the end of (n-1)-th row, forming a continuous string.

Crossrefs

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

Sequence in context: A050072 A267576 A156707 * A267208 A106510 A163806

Adjacent sequences: A131306 A131307 A131308 * A131310 A131311 A131312

Keyword

nonn,tabf,more,uned

Author

Gary W. Adamson, Jun 27 2007

Status

approved