# [seqfan] Re: a permutation of the naturals

Andrew Weimholt andrew.weimholt at gmail.com
Wed Oct 28 19:50:17 CET 2009

```I will submit the sequences tonight, along with one more related sequence.
Each number (assuming there is no number catapulted an infinite number
of times) will eventually catapult one and only one other number, so
we can define another sequence in which a(n) is the number that n
catapults in the generation of the first sequence.

I will also incorporate Benoit's suggestion of beginning with 0, which
does not affect the
rest of the sequence.

One interesting observation, when collecting more terms: 29 is
catapulted 17 times and ends up in a position offset by more than
150000 from its original position.
Also, 0,1,54 are the only numbers I found so far with a(n)=n.

Andrew

On 10/28/09, Benoît Jubin <benoit.jubin at gmail.com> wrote:
> Interesting sequence, I'm looking forward to seeing its graph.
>
>  I suggest you begin the sequence at 0 since it does not require any
>  special treatment.
>
>  Benoit
>
>
>
>  On Wed, Oct 28, 2009 at 10:10 AM, Andrew Weimholt
>  <andrew.weimholt at gmail.com> wrote:
>  > This idea was inspired by one of Eric Angelini's recent posts (angry numbers).
>  >
>  > We start with the natural numbers in their normal positions,
>  > and then the number in position 1 (which happens to be 1), catapults
>  > the number to its right to a position 1 further to the right.
>  > So after the first step, we have 1,3,2,4,5,6,7,8...
>  > Then the number now in position 2, (which is 3), catapults the number
>  > to its right (which is 2) to a position 3 further to the right
>  > Now we have, 1,3,4,5,6,2,7,8...
>  > In the nth step, the number now in the nth position (which will be
>  > a(n)) catapults the number to its right to a position a(n) further to
>  > the right.
>  >
>  > The sequence, beginning at n=1 is...
>  >
>  > 1, 3, 4, 6, 7, 5, 10, 2, 13, 12, 14, 16, 18, 19, 21,
>  > 23, 8, 25, 15, 28, 17, 24, 32, 33, 20, 36, 22, 38, 40, 41, 42,
>  > 44, 45, 47, 31, 35, 50, 52, 27, 55, 11, 58, 59, 61, 63, 64, 66,
>  >
>  > The inverse permutation is...
>  >
>  > 1, 8, 2, 3, 6, 4, 5, 17, 152, 7, 41, 10, 9, 11, 19,
>  > 12, 21, 13, 14, 25, 15, 27, 16, 22, 18, 57, 39, 20,
>  >
>  > The following sequence gives the number of times n is catapulted
>  >
>  > 0, 3, 0, 0, 1, 0, 0, 2, 6, 0, 3, 1, 0, 0, 1,
>  > 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 2, 2, 0,
>  >
>  > Not sure these are worth submitting, but thought I'd at least share
>  > them the seqfan list
>  >
>  > Andrew
>  >
>  >
>
> > _______________________________________________
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>
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```

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