A133404 Table of sum of numerator and denominator of Farey sequences, read by rows.

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, 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, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 11, 13, 15, 2

Offset: 1

Author

Jonathan Vos Post, Nov 24 2007

Keywords

• easy
• nonn
• tabf

Comments

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).

Examples

			F(1) = (0/1, 1/1) -> (0+1=1, 1+1=2).
F(2) = (0/1, 1/2, 1/1) -> (0+1=1, 1+2=3, 1+1=2).
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).
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).
The 5th row is formed from the 5th row of the table of Farey fractions:
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).
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).
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).
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).
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005728, A007305, A007306, A049448.

Programs

Maple

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:
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

Mathematica

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 *)

Formula

A007305/A007306 maps to A007305+A007306 as shown in examples.

Extensions

a(17) inserted by Nathaniel Johnston, Apr 27 2011