# help a new sequence

Dion Gijswijt gijswijt at science.uva.nl
Tue Jun 8 09:31:14 CEST 2004

```Hi Murty,

If I remember correctly, the sequence stops at 42 because then both

But maybe there does exist a permutation \Pi of the natural numbers such
that \Pi(1)=1, |\Pi(n+1)-\Pi(n)|=n+1 for n=1,2,...
I couldn't find one.

--Dion Gijswijt

murthy amarnath wrote:
> Neil and seq fans,
> what shape the following sequence would  finally take?
> Rearrangement of natural numbers.
> a(1) = 1 and let n = a(k) then  n+1= a(k-n-1) in case
> a(k-n-1)is not occupied by some non zero number else
> n+1 = a(k+n+1).
>
> One can start building up the sequence like this:
> Place zeros at locations which can not be decided in
> that step and then replace them by nonzero numbers as
> and when possible in further steps ;
>  first step: 1,0,2,0,0,3,...
>  in the second step the zero following 1 is replaced
> by 4 etc.
> --1,4,2,0,0,3,5,0,0,0,0,0,6,
> --1,4,2,0,0,3,5,0,0,0,0,0,6,0,0,0,0,0,0,7,
> --1,4,2,0,0,3,5,0,0,0,0,8,6,0,0,0,0,0,0,7
> --1,4,2,0,0,3,5,0,0,0,0,8,6,0,0,0,0,0,0,7,9,
> --1,4,2,0,0,3,5,0,0,0,10,8,6,0,0,0,0,0,0,7,9,
> --1,4,2,0,0,3,5,0,0,0,10,8,6,0,0,0,0,0,0,7,9,11,
> --1,4,2,0,0,3,5,0,0,12,10,8,6,0,0,0,0,0,0,7,9,11,
> --1,4,2,0,0,3,5,0,0,12,10,8,6,0,0,0,0,0,0,7,9,11,13,
> --1,4,2,0,0,3,5,0,14,12,10,8,6,0,0,0,0,0,0,7,9,11,13,
> --1,4,2,0,0,3,5,0,14,12,10,8,6,0,0,0,0,0,0,7,9,11,13,15
> --1,4,2,0,0,3,5,16,14,12,10,8,6,0,0,0,0,0,0,7,9,11,13,15
> --1,4,2,0,0,3,5,16,14,12,10,8,6,0,0,0,0,0,0,7,9,11,13,15,17
>
> --1,4,2,0,0,3,5,16,14,12,10,8,6,0,0,0,0,0,0,7,9,11,13,15,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,...
>
> and so on
> Conjecture:There will remain no zeros left finally.
> Can some one please let me know what is a(4) and a(5).
> thanks
> regards
> amarnath murthy
>
>
>
>
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```