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A177405 Form triangle of weighted Farey fractions; read numerators by rows. 6
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 (list; graph; refs; listen; history; text; internal format)
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: A294599 A080940 A080941 * A323302 A303903 A209318

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

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Last modified November 18 17:57 EST 2019. Contains 329288 sequences. (Running on oeis4.)