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

Author

Wouter Meeussen and William P. Orrick, Mar 03 2024

Keywords

• nonn
• tabf

Comments

Length of row n is A370164(n).

Examples

			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).
		

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

• 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].

Crossrefs

Markov numbers: A002559.

Markov tree: A327345, A368546.

Cf. A370164.

Programs

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)]