Markov tree-- Markov numberマルコフ数

There are two simple ways to obtain a new Markov triple from an old one (x, y, z). First, one may permute the 3 numbers x,y,z, so in particular one can normalize the triples so that x ≤ y ≤ z. Second, if (x, y, z) is a Markov triple then so is (x, y, 3xy − z). Applying this operation twice returns the same triple one started with. Joining each normalized Markov triple to the 1, 2, or 3 normalized triples one can obtain from this gives a graph starting from (1,1,1) as in the diagram. This graph is connected; in other words every Markov triple can be connected to (1,1,1) by a sequence of these operations. If one starts, as an example, with (1, 5, 13) we get its three neighbors (5, 13, 194), (1, 13, 34) and (1, 2, 5) in the Markov tree if z is set to 1, 5 and 13, respectively. For instance, starting with (1, 1, 2) and trading y and z before each iteration of the transform lists Markov triples with Fibonacci numbers. Starting with that same triplet and trading x and z before each iteration gives the triples with "Pell numbers↓ '".
< https://en.wikipedia.org/wiki/Markov_number >
(< https://shinichiwanko2000.livedoor.blog/archives/28594745.html >2025年05月25日 double Bruhat cells(Examples )---cluster algebraクラスター代数 )


Both the Pell numbers and the companion Pell numbers may be calculated by means of a recurrence relation similar to that for the Fibonacci numbers, and both sequences of numbers grow exponentially, proportionally to powers{←,　(1 ± √2)^n ←←(Computations and connections---/wiki/Pell_number ) } of the silver ratio 1 + √2[< https://tadashityutyu.blogspot.com/2025/04/silver-ratio.html >silver ratio白銀比(無理数-比?!なのでﾅｶﾅｶむずかしいカナ！♪・？？・　・ )4月 22, 2025(The quadratic formula gives the two solutions 1 ± √2 , the decimal expansion of the positive root begins as ⁠ 2.414 213 562 373 095... (sequence A014176< https://oeis.org/A014176 > in the OEIS ). ), < https://shinichiwanko2000.livedoor.blog/archives/28578806.html >2025年05月23日 silver ratio白銀比{a sleeve袂ﾀﾓﾄが、x^2-2x-1, thenだから・this moment pointこう？・なってしまうsee ifカナ？？・? ?,, } ]. As well as being used to approximate the square root of two, Pell numbers can be used to find square triangular numbers, to construct integer approximations to the right isosceles triangle, and to solve certain combinatorial enumeration problems.
< https://en.wikipedia.org/wiki/Pell_number >
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(relevant?, )
https://plaza.rakuten.co.jp/tadashityutyu/diary/201807250000/
2018年07月25日Pell numbers---Pell numberペル数
The Pell numbers are defined by the recurrence relation循環関係{？は (P_{n-1}+P_{n}) /P_{n} =[≒? ] 1 /1, 3 /2, 7 /5, 17 /12, 41 /29, 99 /70, . .. とだんだん√2の値に近付く.↑↑ }
e tc??< https://cse.google.com/cse?ie=UTF-8&q=Pell+number+site%3Aplaza.rakuten.co.jp%2Ftadashityutyu&cx=002636997843861491696%3Av3cnak5qsac&siteurl=plaza.rakuten.co.jp%2Ftadashityutyu%2F&ref=my.plaza.rakuten.co.jp%2Fdiary%2Flist%2F3%2F%3Fmonth%3D201712&ss= >Pell number site:plaza.rakuten.co.jp/tadashityutyu