|
%I #56 Feb 06 2014 19:26:18
%S 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,
%T 65,33,44,55,48,14,50,40,32,60,11,61,1860,341,541,146340,15447,20596,
%U 25745,32208,2540,1524,635,381,508,16125,4515,936,75,72,54,90,56
%N 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).
%C 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. For example, starting with 6 we get 6,8,10,24,7,25,15,9,12,5,3,4, stop (A235599).
%C By beginning with 3 or 4, we make sure that the 5,3,4 dead-end is never available.
%C If infinite, is it a permutation of the integers >= 3? This seems likely. Proving it doesn't seem easy though.
%C Comment from _Jim Nastos_, Dec 30 2013: Your question about whether the sequence can 'paint itself into a corner' is essentially asking if the Pythagorean graph has a Hamiltonian path. As far as I know, the questions in the Cooper-Poirel paper (see link) are still unanswered. They ask 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 integers >=3).
%C Lars Blomberg has computed the sequence out to 3 million terms without finding a dead end.
%C Position of k>2: 0, 1, 2, 7, 10, 6, 4, 8, 35, 3, 67, 30, 5, 13, 89, 15, 143, 12, 22, 118, 385, 9, 11, ..., see A236243. - _Robert G. Wilson v_, Jan 17 2014
%H Robert G. Wilson v, <a href="/A235598/b235598.txt">Table of n, a(n) for n = 0..10000</a>
%H J. Cooper and C. Poirel, <a href="http://www.math.sc.edu/~cooper/pth.pdf">Note on the Pythagorean Triple System</a>
%t f[s_List] := Block[{n = s[[-1]]}, sol = Solve[ x^2 + y^2 == z^2 && x > 0 && y > 0 && z > 0 && (x == n || z == n), {x, y, z}, Integers]; Append[s, Min[ Complement[ Union[ Extract[sol, Position[ sol, _Integer]]], s]]]]; lst = Nest[f, {3}, 250] (* _Robert G. Wilson v_, Jan 17 2014 *)
%Y Cf. A020882, A020883, A020884, A235599, A236243, A236244.
%K nonn,nice
%O 0,1
%A _Jack Brennen_, Dec 26 2013
|
