0,1

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 (1 + 3 + 8) zeros and (1 + 2 + 5 + 13) = 21 ones; with the ratio of ones to zeros tending to phi^2 = 2.618...

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.

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.

Cf. A027941, A001906.

Sequence in context: A080909 A087755 A050072 this_sequence A106510 A145362 A075437

Adjacent sequences: A131306 A131307 A131308 this_sequence A131310 A131311 A131312

nonn

Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 27 2007