A177405 Form triangle of weighted Farey fractions; read numerators by rows.

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

Offset

0,5

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.

James Propp asks: Does every fraction between 0 and 1 with odd denominator appear in the triangle?

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, 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
0 1 .2 1 .4 .5 2 .5 .4 1 2 3 4 13 14 5 4 3 2 .9 12 5 14 13 4 .9 6 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

Mathematica

Mma code from James Propp:
        Lengthen[L_] :=
         Module[{i, M}, M = Table[0, {3 Length[L]}];
          M[[1]] = Numerator[L[[1]]]/(2 + Denominator[L[[1]]]);
          M[[2]] = 2*Numerator[L[[1]]]/(1 + 2 Denominator[L[[1]]]);
          For[i = 1, i < Length[L], i++, M[[3 i]] = L[[i]];
           M[[3 i + 1]] = (2 Numerator[L[[i]]] +
               Numerator[L[[i + 1]]])/(2 Denominator[L[[i]]] +
               Denominator[L[[i + 1]]]);
           M[[3 i + 2]] = (Numerator[L[[i]]] +
               2 Numerator[L[[i + 1]]])/(Denominator[L[[i]]] +
               2 Denominator[L[[i + 1]]])]; M[[3 Length[L]]] = L[[Length[L]]];
           Return[M]]
        WF[n_] := WF[n] = If[n == 0, {1}, Lengthen[WF[n - 1]]]

Crossrefs

Cf. A177407, A177903, A006842/A006843.

Sequence in context: A080940 A080941 A346705 * A362421 A323302 A303903

Adjacent sequences: A177402 A177403 A177404 * A177406 A177407 A177408

Keyword

nonn,frac,tabf,easy

Author

N. J. A. Sloane, Dec 10 2010

Extensions

a(45)-a(80) and some corrected terms from Nathaniel Johnston, Apr 12 2011

Status

approved