OFFSET
1,8
COMMENTS
Algorithm:
1. Place a dot, representing p_1, on the unit circle at (1,0), or equivalently, at angle 0 degrees, and label this point as the complex number lambda_1 = 1 = e^(i*0).
2. Place a dot, representing p_2, on the unit circle at (-1,0), or equivalently, at angle 180 degrees, and label this point as the complex number lambda_2 = -1 = e^(i*Pi).
3. For n > 2, supposing that dots representing p_1, ..., p_n have already been placed on the unit circle, place a dot, representing p_{n+1}, on the unit circle as follows: Starting at the dot representing p_n, proceed counterclockwise around the circle until you reach a dot representing one of p_1, p_2, ..., p_{n-1}.
4. Starting at this latter dot, proceed counterclockwise until you reach another dot representing one of p_1, p_2, ..., p_n.
5. Continue this process until you have traversed a(n) (of A077218) successive arcs connecting dots.
6. Proceed in a counterclockwise direction one more time to the next dot and place the dot which will represent p_{n+1} at the midpoint of the arc just traversed. Label this point as the complex number lambda_{n+1} = e^(i*theta_{n+1}).
The sequence is of interest since the points seem to cluster in the mentioned arc as well as in the closed arc from 180 degrees to 225 degrees. The sequences have been constructed by James Farrington up to n=256, which suggests the clustering. The precise clustering properties are formulated as questions in a file which is referenced below under LINKS.
LINKS
EXAMPLE
Following this procedure, we would place a dot representing p_3 on the unit circle at (0,-1), or equivalently, at angle 270 degrees, and we would place a dot representing p_4 on the unit circle at the midpoint of the arc connecting the point (-1,0) to the point (0,-1), that is, at (-1/root2,-1/root2), or equivalently, at angle 225 degrees.
Thus lambda_3 = -i = e^(3*Pi/2) and lambda_4 = e^(5*Pi/4).
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Bernard X. Russo, Apr 09 2019
STATUS
approved