[seqfan] Re: Simple sequence based on Pythagorean triples

[Jim Nastos]

Your questions about whether the sequence can 'paint intself in a corner'
is essentially asking if the pyth graph has a hamiltonian path... as far as
I know, the questions in this paper are still unanswered:
http://www.math.sc.edu/~cooper/pth.pdf
  where it is asked whether the graph is k-colorable with a finite k, or
whether it is even connected (sort of equivalent to your question of
whether it is a permutation of the ints>=3).
  If anyone knows whether those questions in the paper have been resolved,
I would be very interested.
J


On Thu, Dec 26, 2013 at 10:48 AM, Jack Brennen <jfb at brennen.net> wrote:

> This sequence seems simple enough and yet is not in the OEIS:
>
> Begin with a(0) = 3.
>
> Let a(n) for n > 0 be the smallest positive integer not yet
> in the sequence which forms part of a Pythagorean triple
> when paired with a(n-1).
>
> I believe that the sequence begins:
>
>   3,4,5,12,9,15,8,6,10,24,7,25,20,16,30,18,80,39,36,27,
>   45,28,21,29,420,65,33,44,55,48,14,50,40,32,60,11,61,
>   1860,341,541,146340,15447,20596,25745,32208,2540,
>   1524,635,381,508,16125,4515,936,75,72,54,90,56,42,
>   58,840,41,841,580,68,51,85,13,84,35,37,684,285,152,
>   114,190,336,52,165,88,66,110,96,100,105,63,87,116,
>   145,17,144,108,81,135,153,104,78,130,112,113,6384,
>   640,312,91,109,5940,567,540,57,76,95,168,26,170,102,
>   136,64,120,22,122,3720,682,1082,292680,30894,41192,
>   51490,64416,2513,8616,3590,2154,2872,5385,3231,4308,
>   1795,1077,1436,128877,12920,663,180,19,...
>
> (Any typos are part of my cut-and-paste...)
>
> Two questions:
>
> Is the sequence infinite?  Can it "paint itself into
> a corner" at any point?  Note that picking any starting
> point >= 5 seems to lead to a finite sequence ending in
> 5,3,4:
>
>    6,8,10,24,7,25,15,9,12,5,3,4 stop
>
> By beginning with 3 or 4, you make sure that the 5,3,4
> dead-end is never available.
>
>
> If infinite, is it a permutation of the integers >= 3?
> It seems likely.  Proving it doesn't seem easy though.
>
>
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