cp's OEIS Frontend

The rule can be generalized for any number of starting terms s: (xs, ..., x2, x1) -> (x1, x1 + x2, ..., x1 + x2 + ... + xs), using (0, 1, ..., s-1) as seed values. This sequence is s=3, and s=2 yields the Fibonacci series.

For s=3 the ratio of S1 (the first in the sub-series) to S3 (the 3rd in the sub-series) converges on 2.2469796 and the ration of S2 (the 2nd in the sub-series) to S3 converges on 1.2469796 thus the difference, S2-S3, converges on 1 regardless of the seed values used.

For s=20 the series is: 19, 190, 2660, 33915, 445949, ....

a(n-2) is the top left entry of the n-th power of the 3 X 3 matrix [0, 1, 1; 1, 1, 1; 1, 0, 1] or of the 3 X 3 matrix [0, 1, 1; 1, 1, 0; 1, 1, 1]. - R. J. Mathar, Feb 03 2014

From Andrew Pharo, Jun 02 2014: (Start)

For s=2 the ratio of successive terms is 1.6180339887... or phi (or phi(2));

for s=3 this ratio is 2.24697960412319..., phi(3) = 4*cos(Pi/7)^2-1 (see Falbo link);

for s=4 this ratio is 3.5133370918694...;

for s=20 this ratio is 13.0538985560545... and so on.

We can define a function phi(s) which approximates to:

phi(s) ~ phi(2) + theta*(s-2) where theta ~ 0.636264133.

(End)