A370852 Irregular triangle read by rows: row n is the list of residues mod n that occur among the Markov numbers

0, 0, 1, 1, 2, 1, 2, 0, 1, 2, 3, 4, 1, 2, 4, 5, 1, 2, 5, 6, 1, 2, 5, 1, 2, 4, 5, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 4, 5, 6, 7, 9, 10, 1, 2, 5, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 5, 6, 8, 9, 12, 13, 1, 2, 4, 5, 7, 8, 10, 11, 13, 14

Offset

1,5

Comments

Length of row n is A370164(n).

References

Martin Aigner, Markov's theorem and 100 years of the uniqueness conjecture. A mathematical journey from irrational numbers to perfect matchings. Springer, 2013. x+257 pp. ISBN: 978-3-319-00887-5; 978-3-319-00888-2 MR3098784.

Links

Table of n, a(n) for n=1..82.

Martin Aigner, Markov's theorem and 100 years of the uniqueness conjecture. A mathematical journey from irrational numbers to perfect matchings, [archive.org copy of the book].

Example

The first rows are:
n
1:   0
2:   0  1
3:   1  2
4:   1  2
5:   0  1  2  3  4
6:   1  2  4  5
7:   1  2  5  6
8:   1  2  5
9:   1  2  4  5  7  8
10:  0  1  2  3  4  5  6  7  8  9
11:  1  2  4  5  6  7  9 10
12:  1  2  5 10
13:  0  1  2  3  4  5  6  7  8  9 10 11 12
14:  1  2  5  6  8  9 12 13
15:  1  2  4  5  7  8 10 11 13 14
16:  1  2  5  9 13
17:  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16
18:  1  2  4  5  7  8 10 11 13 14 16 17
19:  1  2  3  4  5  6  8  9 10 11 13 14 15 16 17 18
20:  1  2  5  6  9 10 13 14 17 18
For n = 14 residues congruent to 0, 3, or 4 mod 7 are forbidden. (See comments to A370164 for explanation.) All other residues occur. For example, the Markov numbers 1, 2, 5, 34, 610, 1325, 194, and 13 produce the residues shown in row 14 of the triangle (mod 14).

Prog

(SageMath)
def generateAllMarkovTreeResidues(n):
    row = [[1 % n, 5 % n, 2 % n]]
    residuesFound = []
    triplesFound = []
    while row != []:
        newRow = []
        for trpl in row:
            if trpl[1] not in residuesFound:
                residuesFound.append(trpl[1])
            if trpl[2] < trpl[0]:
                trpl.reverse()
            if trpl not in triplesFound:
                triplesFound.append(trpl)
                newRow.append([trpl[0], (3*trpl[0]*trpl[1]-trpl[2]) % n, trpl[1]])
                newRow.append([trpl[1], (3*trpl[1]*trpl[2]-trpl[0]) % n, trpl[2]])
        row = newRow
    residuesFound.sort()
    return(residuesFound)
[r for n in range(1, 16) for r in generateAllMarkovTreeResidues(n)]

Crossrefs

Markov numbers: A002559.

Markov tree: A327345, A368546.

Cf. A370164.

Sequence in context: A211359 A211357 A238416 * A063574 A372095 A144515

Adjacent sequences: A370849 A370850 A370851 * A370853 A370854 A370855

Keyword

nonn,tabf

Author

Wouter Meeussen and William P. Orrick, Mar 03 2024

Status

approved