Search: a236244 -id:a236244
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A235598
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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).
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+10
7
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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
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OFFSET
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0,1
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COMMENTS
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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
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LINKS
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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STATUS
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approved
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0, 1, 2, 7, 10, 6, 4, 8, 35, 3, 67, 30, 5, 13, 89, 15, 143, 12, 22, 118, 385, 9, 11, 112, 19, 21, 23, 14, 355, 33, 26, 247, 69, 18, 70, 188, 17, 32, 61, 58, 888, 27, 20, 403, 871, 29, 250, 31, 65, 77, 1254, 55, 28, 57, 108, 59, 823, 34, 36, 633, 85, 116, 25, 80, 1710, 64, 238, 151, 1202, 54, 1677, 152, 53
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OFFSET
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3,3
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COMMENTS
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The terms of this sequence are conjectured to be a permutation of the natural numbers.
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LINKS
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MATHEMATICA
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(* run the Mmca in A235598 *); Flatten[ Table[ Position[ lst, n, 1, 1], {n, 3, 75}] - 1]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A239431
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Consider the sequence A235598. Recalling that A235598(n) forms part of a Pythagorean triple, a(n) states its relationship to both a(n-1) and a(n+1). 1 denotes the lesser leg, 2 denotes the greater leg and 3 denotes the hypotenuse. The tens place returns its relationship to the side to its left, a(n-1), and the units place its relationship to the side to its right, a(n+1). a(0)=1.
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+10
1
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1, 22, 31, 22, 11, 32, 12, 11, 31, 22, 11, 33, 23, 21, 23, 11, 22, 13, 22, 11, 32, 12, 12, 31, 22, 13, 11, 22, 32, 12, 11, 33, 23, 21, 22, 11, 31, 22, 11, 31, 22, 11, 22, 31, 22, 13, 12, 13, 11, 21, 23, 12, 12, 13, 22, 11, 32, 12, 12, 31, 22, 11, 33, 13, 12, 11, 33, 11, 22, 12, 31, 22, 12, 12, 11, 31, 22, 11, 22, 12
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OFFSET
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0,2
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COMMENTS
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Using the data that is available from Lars Blomberg, and the nine possible arrangements, (a), of the three sides, here are those counts for the first x terms not including a(0):
\x. 10 100 1000 10000 100000 1000000 3000000 aprx percentage.
a\
11:. 3. 21. 164. 1502. 13734. 134087. 401166 ~13.3%
12:. 1. 18. 215. 2120. 21457. 208304. 621859 ~20.7%
13:. 0.. 6.. 94.. 921.. 8884.. 86286. 256802. ~8.5%
21:. 0.. 3.. 51.. 550.. 5120.. 51732. 156588. ~5.2%
22:. 3. 22. 207. 2025. 21013. 214855. 646185 ~21.6%
23:. 0.. 5.. 55.. 657.. 6347.. 64480. 194775. ~6.5%
31:. 2. 12.. 95.. 881.. 8697.. 83631. 249413. ~8.3%
32:. 1.. 7.. 51.. 390.. 4219.. 42112. 126410. ~4.2%
33:. 0.. 6.. 68.. 954. 10529. 114513. 346802 ~11.7%
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LINKS
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EXAMPLE
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a(2)=31 because 5 is the hypotenuse in the 3-4-5 Pythagorean triple, a(n-1) is 4 and 5 is the lesser side in the 5-12-13 Pythagorean triple, a(n+1) is 12.
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MATHEMATICA
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lst={ (* the terms from A235598 *) }; g[j_, k_] := Block[{hyp = Sqrt[ j^2 + k^2], lg = Abs@ Sqrt[ j^2 - k^2]}, If[ IntegerQ@ hyp, If[ Min[j, k] == k, 1, 2], If[ Max[j, k] == k, 3, If[lg > k, 1, 2]]]]; f[n_] := Block[{s = Take[lst, {n - 1, n + 1}]}, 10g[ s[[1]], s[[2]] ] + g[ s[[3]], s[[2]] ]]; f[1] = 1; Array[f, 80]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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