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A235598 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). 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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).

By beginning with 3 or 4, we make sure that the 5,3,4 dead-end is never available.

If infinite, is it a permutation of the integers >= 3? This seems likely.  Proving it doesn't seem easy though.

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).

Lars Blomberg has computed the sequence out to 3 million terms without finding a dead end.

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

LINKS

Robert G. Wilson v, Table of n, a(n) for n = 0..10000

J. Cooper and C. Poirel, Note on the Pythagorean Triple System

MATHEMATICA

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 *)

CROSSREFS

Cf. A020882, A020883, A020884, A235599, A236243, A236244.

Sequence in context: A302752 A046964 A296966 * A191197 A055493 A109350

Adjacent sequences:  A235595 A235596 A235597 * A235599 A235600 A235601

KEYWORD

nonn,nice

AUTHOR

Jack Brennen, Dec 26 2013

STATUS

approved

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Last modified May 16 20:11 EDT 2022. Contains 353720 sequences. (Running on oeis4.)