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A133404 Table of sum of numerator and denominator of Farey sequences, read by rows. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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).

LINKS

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

FORMULA

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

EXAMPLE

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).

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 *)

CROSSREFS

Cf. A005728, A007305, A007306, A049448.

Sequence in context: A118851 A112383 A272464 * A134627 A064881 A131967

Adjacent sequences:  A133401 A133402 A133403 * A133405 A133406 A133407

KEYWORD

easy,nonn,tabf

AUTHOR

Jonathan Vos Post, Nov 24 2007

EXTENSIONS

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

STATUS

approved

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Last modified July 8 05:33 EDT 2020. Contains 335513 sequences. (Running on oeis4.)