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

%I A177405 #34 May 07 2018 03:37:48
%S A177405 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,
%T A177405 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,
%U A177405 12,5,14,13,4,9,6,1,4,5,2,7,8,3,10,11
%N A177405 Form triangle of weighted Farey fractions; read numerators by rows.
%C A177405 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.
%C A177405 James Propp asks: Does every fraction between 0 and 1 with odd denominator appear in the triangle?
%D A177405 James Propp, Posting to the Math Fun Mailing List, Dec 10 2010.
%H A177405 Nathaniel Johnston, <a href="/A177405/b177405.txt">Table of n, a(n) for n = 0..29533</a> (first 10 rows of triangle)
%H A177405 Dhroova Aiylam, Tanya Khovanova, <a href="https://arxiv.org/abs/1711.01475">Weighted Mediants and Fractals</a>, arXiv:1711.01475 [math.NT], 2017.
%e A177405 Triangle begins:
%e A177405 0 1
%e A177405 - -
%e A177405 1 1
%e A177405 0 1 2 1
%e A177405 - - - -
%e A177405 1 3 3 1
%e A177405 0 1 2 1 4 5 2 5 4 1
%e A177405 - - - - - - - - - -
%e A177405 1 5 7 3 9 9 3 7 5 1
%e A177405 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
%e A177405 - - -- - -- -- - -- -- - - - - -- -- - - - - -- -- - -- -- - -- - -
%e A177405 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
%t A177405 Mma code from _James Propp_:
%t A177405         Lengthen[L_] :=
%t A177405          Module[{i, M}, M = Table[0, {3 Length[L]}];
%t A177405           M[[1]] = Numerator[L[[1]]]/(2 + Denominator[L[[1]]]);
%t A177405           M[[2]] = 2*Numerator[L[[1]]]/(1 + 2 Denominator[L[[1]]]);
%t A177405           For[i = 1, i < Length[L], i++, M[[3 i]] = L[[i]];
%t A177405            M[[3 i + 1]] = (2 Numerator[L[[i]]] +
%t A177405                Numerator[L[[i + 1]]])/(2 Denominator[L[[i]]] +
%t A177405                Denominator[L[[i + 1]]]);
%t A177405            M[[3 i + 2]] = (Numerator[L[[i]]] +
%t A177405                2 Numerator[L[[i + 1]]])/(Denominator[L[[i]]] +
%t A177405                2 Denominator[L[[i + 1]]])]; M[[3 Length[L]]] = L[[Length[L]]];
%t A177405            Return[M]]
%t A177405         WF[n_] := WF[n] = If[n == 0, {1}, Lengthen[WF[n - 1]]]
%Y A177405 Cf. A177407, A177903, A006842/A006843.
%K A177405 nonn,frac,tabf,easy
%O A177405 0,5
%A A177405 _N. J. A. Sloane_, Dec 10 2010
%E A177405 a(45)-a(80) and some corrected terms from _Nathaniel Johnston_, Apr 12 2011