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

Author

N. J. A. Sloane, Dec 10 2010

Keywords

• nonn
• frac
• tabf
• easy

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?

Examples

			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
		

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.

Crossrefs

Cf. A177407, A177903, A006842/A006843.

Programs

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]]]

Extensions

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