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A370852
Irregular triangle read by rows: row n is the list of residues mod n that occur among the Markov numbers
1
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.
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
KEYWORD
nonn,tabf
AUTHOR
STATUS
approved