Re: Guess the Permutation

From: Jyrki Lahtonen (lahtonen_at_utu.fi)
Date: 01/12/05 

Date: Wed, 12 Jan 2005 12:35:30 +0200

Leroy Quet wrote:
> Here are the first terms of a sequence which
> forms a permutation of the positive integers.
>
> 1,2,3,6,4,9,5,12,7,16,8,19,10,23,11,26,
> 13,30,14,33,15,36,17,...
>
> Try to guess the rule I used to generate this permutation.
> (Not in EIS.)
>

I dunno. The first and most obvious observation is that
the entries in the odd-numbered positions are the smallest
naturals that haven't appeared earlier in the sequence.

The entries in the even-numbered positions seem to be gotten
by applying a function of the type f(x)=2x+n to the
preceeding entry. The constant terms n keep increasing,
but not very regularly, i.e. when getting the second and the
fourth entries from the first (resp. the third) the value
n=0 is used. Then n=1 is also used twice to get the sixth
and the eighth entries. At this point I thought I have it,
but the value n=2 is used only once (when computing that
7 must be followed by 16=2*7+2), then n=3, and n=4 are both
used twice, but n=5 is again only used once, and n=6 is
used once (to get the step from 15 to 36).

So this fits almost but not quite. I rather suspect that
the above observations are consequences of the rule rather
than parts of the rule themselves. I guess I could complete
this into a rule of the type that n=n(i)=floor(k*i+b) for
suitable constants k and b (the index i indicates that this
value for n is used when computing the entry at position 2i.
However, that would be too ugly, and the constants k and
b wouldn't be uniquely determined from the sample data given.

More work, but if you feel like making the rule public
I won't object:)

Cheers,

Jyrki Lahtonen, Turku, Finland

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