A133404 - OEIS

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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 (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: A364603 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 August 18 23:41 EDT 2024. Contains 375284 sequences. (Running on oeis4.)