cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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).

Original entry on oeis.org

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
Offset: 0

Views

Author

Jack Brennen, Dec 26 2013

Keywords

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

Crossrefs

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

Programs

A236244 Records in A235598.

Original entry on oeis.org

3, 4, 5, 12, 15, 24, 25, 30, 80, 420, 1860, 146340, 292680, 3031080, 6355440, 202025100, 404050200, 674215920, 1309133280, 1759047480, 3141424023360, 6282848046720, 19013977277880, 2604166475512980, 295127694592600800, 1528966300926585360
Offset: 1

Views

Author

Robert G. Wilson v, Jan 22 2014

Keywords

Comments

These occur at positions: 0, 1, 2, 3, 5, 9, 11, 14, 16, 24, 37, 40, 123, 639, 926, 1040, 2263, 4266, 9242, 13783, 19157, 128911, 133369, 165603, 279520, 367234, …, .

Crossrefs

Cf. A235598, A236243.

Programs

  • Mathematica
    (* run the Mmca in A235598 *); k = 1; lsu = {}; mx = 0; While[k < 435000, If[lst[[k]] > mx, mx = lst[[k]]; Print[mx]; AppendTo[lsu, mx]]; k++]; lsu

A239431 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.

Original entry on oeis.org

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
Offset: 0

Views

Author

Robert G. Wilson v, Mar 20 2014

Keywords

Comments

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%

Examples

			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.

Links

Crossrefs

Cf. A235598, A236243, A236244.

Programs

  • Mathematica
    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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