[seqfan] Re: Turtle transform

[M. F. Hasler]

This is (unless I'm wrong) exactly (or up to a sign, maybe)
what I defined to be the A1477 <https://oeis.org/A001477>-turtle transform
of A40 <https://oeis.org/A000040>.

Maximilian

On Tue, Dec 10, 2013 at 11:25 AM, Neil Fernandez <primeness at borve.org>wrote:

> 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
>
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