A177407 Form triangle of weighted Farey fractions; read denominators by rows.

1, 1, 1, 3, 3, 1, 1, 5, 7, 3, 9, 9, 3, 7, 5, 1, 1, 7, 11, 5, 17, 19, 7, 17, 13, 3, 5, 7, 9, 27, 27, 9, 7, 5, 3, 13, 17, 7, 19, 17, 5, 11, 7, 1, 1, 9, 15, 7, 25, 29, 11, 27, 21, 5, 9, 13, 17, 53, 55, 19, 15, 11, 7, 31, 41, 17, 47, 43, 13, 29, 19, 3, 11, 13, 5, 17, 19, 7, 23, 25, 9

Offset

0,4

Comments

Start with the list of fractions 0/1, 1/1 and repeatedly insert the weighted mediants (2a+c)/(2b+d) and (a+2c)/(b+2d) between every pair of adjacent elements a/b and c/d of the list. The fractions are to be reduced before the insertion step.

References

James Propp, Posting to the Math Fun Mailing List, Dec 10 2010.

Links

Nathaniel Johnston, Table of n, a(n) for n = 0..29533 (first 10 rows of triangle)

Dhroova Aiylam and Tanya Khovanova, Weighted Mediants and Fractals, arXiv:1711.01475 [math.NT], 2017.

Example

Triangle begins:
  0 1
  - -
  1 1
.
  0 1 2 1
  - - - -
  1 3 3 1
.
  0 1 2 1 4 5 2 5 4 1
  - - - - - - - - - -
  1 5 7 3 9 9 3 7 5 1

Crossrefs

Cf. A177405, A177903, A006842/A006843.

Sequence in context: A220837 A187371 A056611 * A135669 A132729 A196493

Adjacent sequences: A177404 A177405 A177406 * A177408 A177409 A177410

Keyword

nonn,frac,tabf,easy

Author

N. J. A. Sloane, Dec 10 2010

Extensions

a(44)-a(80) and some corrected terms from Nathaniel Johnston, Apr 12 2011

Status

approved