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%I #16 Mar 02 2014 19:45:51
%S 1,2,1,3,2,1,4,3,5,2,1,5,4,3,5,7,2,1,6,5,4,7,3,8,5,7,9,2,1,7,6,5,4,7,
%T 3,8,5,7,9,11,2,1,8,7,6,5,9,4,7,10,3,11,8,5,12,7,9,11,13,2,1,9,8,7,6,
%U 5,9,4,11,7,10,3,11,8,13,5,12,7,9,11,13,15,2
%N Table of sum of numerator and denominator of Farey sequences, read by rows.
%C Start with the Farey sequence F(n) of order n which is the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size. Each row begins with the sum 1 from {0/1}. Each row ends with the sum 2 from {1/1}. The number of elements of the n-th row is A005728(n).
%H Nathaniel Johnston, <a href="/A133404/b133404.txt">Table of n, a(n) for n = 1..10000</a>
%F A007305/A007306 maps to A007305+A007306 as shown in examples.
%e F(1) = (0/1, 1/1) -> (0+1=1, 1+1=2).
%e F(2) = (0/1, 1/2, 1/1) -> (0+1=1, 1+2=3, 1+1=2).
%e F(3) = (0/1, 1/3, 1/2, 2/3, 1/1) -> (0+1=1, 1+3=4, 1+2=3, 2+3=5, 1+1=2).
%e F(4) = (0/1, 1/4, 1/3, 1/2, 2/3, 3/4, 1/1) -> (0+1=1, 1+4=5, 1+3=4, 1+2=3, 2+3=5, 3+4=7, 1+1=2).
%e The 5th row is formed from the 5th row of the table of Farey fractions:
%e F(5) = (0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1) whose sum of numerators and denominators is (1, 6, 5, 4, 7, 3, 8, 5, 7, 9, 2).
%e F(6) = (0/1, 1/6, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 5/6, 1/1} whose sums are (1, 7, 6, 5, 4, 7, 3, 8, 5, 7, 9, 11, 2).
%e F(7) = (0/1, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 2/5, 3/7, 1/2, 4/7, 3/5, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 1/1) whose sums are (1, 8, 7, 6, 5, 9, 4, 7, 10, 3, 11, 8, 5, 12, 7, 9, 11, 13, 2).
%e F(8) = (0/1, 1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, 4/7, 3/5, 5/8, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 7/8, 1/1) whose sums are (1, 9, 8, 7, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 11, 13, 15, 2).
%p Farey := proc(n) option remember: local j,s: if(n=1)then return {0,1}: else s:=procname(n-1): for j from 1 to n-1 do s := s union {j/n}: od: fi: end:
%p for n from 1 to 8 do F:=sort(convert(Farey(n),list)): nF:=nops(F): for m from 1 to nF do printf("%d, ",numer(F[m])+denom(F[m])): od: printf("\n"): od: # _Nathaniel Johnston_, Apr 27 2011
%t Farey[n_] := Union[ Flatten[ Join[{0}, Table[a/b, {b, n}, {a, b}]]]]; Table[ Numerator[Farey[n]] + Denominator[Farey[n]], {n, 8}] // Flatten (* _Robert G. Wilson v_, Jun 10 2011 *)
%Y Cf. A005728, A007305, A007306, A049448.
%K easy,nonn,tabf
%O 1,2
%A _Jonathan Vos Post_, Nov 24 2007
%E a(17) inserted by _Nathaniel Johnston_, Apr 27 2011
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