[seqfan] Re: Turtle transform

[Neil Fernandez]

Hi seqfans,

M. F. Hasler <seqfan at hasler.fr> writes

>inspired by a private communication from Eric Angelini,
>I suggest a new(?) [integer] sequence transformations
>(plus some variants), which I tentatively call "Turtle >
>transformation"(s)

William Keith <william.keith at gmail.com> writes:

>I did something like this for primes, although the rule was simply to
>make a [right] turn at node n of the walk if n was a prime. 

Eric Angelini <Eric.Angelini at kntv.be> writes:

>here are a few diagrams -- and ideas.
>Many thanks to Jean-Marc Falcoz and Maximilian Hasler,

Here's a nice transform of the primes. Define a 'click' as a
counterclockwise turn through 90 degrees. Starting at the origin, walk 2
steps in the +ve real direction, turn 1 click, walk 3, turn 2 clicks,
walk 5,..., walk pi(i), turn i clicks,..., where pi(n) is the nth prime.

Defining a(n) as the prime sequence, we get:

b(n) := a(1)  + a(2)*(i) + a(3)*((i^2)*i) + a(4)*((i^3)*(i^2)*i) + ...

= sum from k=1 to n of pi(k)*i^(k(k-1)/2)

Here's a plot for n up to 100, showing only nodes, not paths:

<http:/borve.org/plots/100.jpg>

For n up to 1000, some very nice features appear more clearly:

<http:/borve.org/plots/1000.jpg>

Up to 10000:

<http:/borve.org/plots/10000.jpg>

Neil

-- 
Neil Fernandez