# A Family Of Permutations of the Positive Integers

Leroy Quet qq-quet at mindspring.com
Sat Dec 20 04:18:29 CET 2003

```[also  posted to sci.math]

Let

a(1) = 1;
a(2) = n, n = integer >= 2;

a(m+2)  = the |a(m+1) -a(m)|th highest yet-unpicked positive integer.

By "yet-unpicked",
I mean an integer that is not among {a(1),a(2),...,a(m+1)}.

So, for each n >=2, we get a permutation of the + integers.

Now, the n =2 case is uninteresting, just giving the + integers in their
own order.

But it seems that for all sufficiently large n's, the terms of the
sequence fall into a specific pattern.

And, for all sufficiently high n's, every 3rd term forms a sequence of
constants.

So, if a(n,m) = the m_th term of the sequence with a(n,2) = n (and a(n,1)
= 1),

then, as n -> oo, we can get the sequence:

A(m) = a(n,3m-2),

which begins (perhaps...):

A(m)  ->  1, 2, 5, 6, 9(?),...

Generally, the a-sequence, for all sufficiently high n's,

is(??)..

1, n, n+1, 2, n+3, n+6, 5, n+8, n+11, 6, ...

Anything interesting anyone can add to the knowledge of these
permutations??

thanks,
Leroy
Quet

```