|
Search: id:A119016
|
|
|
| A119016 |
|
Numerators of "Farey fraction" approximations to sqrt(2). |
|
+0
6
|
|
| 1, 0, 1, 2, 3, 4, 7, 10, 17, 24, 41, 58, 99, 140, 239, 338, 577, 816, 1393, 1970, 3363, 4756, 8119, 11482, 19601, 27720, 47321, 66922, 114243, 161564, 275807, 390050, 665857, 941664, 1607521, 2273378, 3880899, 5488420, 9369319, 13250218
(list; graph; listen)
|
|
|
OFFSET
|
0,4
|
|
|
COMMENT
|
"Add" (meaning here to add the numerators and add the denominators, not to add the fractions) 1/0 to 1/1 to make the fraction bigger: 2/1. Now 2/1 is too big, so add 1/1 to make the fraction smaller: 3/2, 4/3. Now 4/3 is too small, so add 3/2 to make the fraction bigger: 7/5, 10/7, ... Because the continued fraction for sqrt(2) is all 2s, it will always take exactly two terms here to switch from a number that's bigger than sqrt(2) to one that's less. a(n+2) = A082766(n).
|
|
LINKS
|
Dave Rusin, Farey fractions on sci.math
|
|
FORMULA
|
a(0) = 1, a(1) = 0, a(2n) = a(2n-1) + a(2n-2), a(2n+1) = a(2n) + a(2n-2)
|
|
EXAMPLE
|
The fractions are 1/0, 0/1, 1/1, 2/1, 3/2, 4/3, 7/5, 10/7, 17/12, ...
|
|
CROSSREFS
|
Cf. A097545, A097546 gives the similar sequence for pi. A119014, A119015 gives the similar sequence for e. A002965 gives the denominators for this sequence. A082766 is a(n+2). Also very closely related to A001333, A052542, and A000129.
See A082766 for another version.
Adjacent sequences: A119013 A119014 A119015 this_sequence A119017 A119018 A119019
Sequence in context: A051449 A018143 A136570 this_sequence A082766 A082958 A060166
|
|
KEYWORD
|
easy,frac,nonn
|
|
AUTHOR
|
Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), May 08 2006
|
|
|
Search completed in 0.002 seconds
|
