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A119016 Numerators of "Farey fraction" approximations to sqrt(2). 8
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, 22619537, 31988856 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

"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).

a(2n) are the interleaved values of m such that 2*m^2-2 and 2*m^2+2 are squares, respectively; a(2n+1) are the corresponding integer square roots. - Richard R. Forberg, Aug 19 2013

Apart from the first two terms, this is the sequence of numerators of the convergents of the continued fraction expansion sqrt(2) = 1/(1 - 1/(2 + 1/(1 - 1/(2 + 1/(1 - ....))))). - Peter Bala, Feb 02 2017

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Dave Rusin, Farey fractions on sci.math [Broken link]

Dave Rusin, Farey fractions on sci.math [Cached copy]

Index entries for linear recurrences with constant coefficients, signature (0,2,0,1).

FORMULA

From Joerg Arndt, Feb 14 2012: (Start)

a(0) = 1, a(1) = 0, a(2n) = a(2n-1) + a(2n-2), a(2n+1) = a(2n) + a(2n-2).

G.f.: (1 - x^2 + 2*x^3)/(1 - 2*x^2 - x^4). (End)

a(n) = 1/4*(1-(-1)^n)*(-2+sqrt(2))*(1+sqrt(2))*((1-sqrt(2))^(1/2*(n-1))-(1+sqrt(2))^(1/2*(n-1)))+1/4*(1+(-1)^n)*((1-sqrt(2))^(n/2)+(1+sqrt(2))^(n/2)). - Gerry Martens, Nov 04 2012

EXAMPLE

The fractions are 1/0, 0/1, 1/1, 2/1, 3/2, 4/3, 7/5, 10/7, 17/12, ...

MAPLE

f:= gfun:-rectoproc({a(n+4)=2*a(n+2) +a(n), a(0)=1, a(1)=0, a(2)=1, a(3)=2}, a(n), remember):

map(f, [$0..100]); # Robert Israel, Jun 10 2015

MATHEMATICA

f[x_, n_] := (m = Floor[x]; f0 = {m, m+1/2, m+1}; r = ({a___, b_, c_, d___} /; b < x < c) :> {b, (Numerator[b] + Numerator[c]) / (Denominator[b] + Denominator[c]), c}; Join[{m, m+1}, NestList[# /. r &, f0, n-3][[All, 2]]]); Join[{1, 0 }, f[Sqrt[2], 38]] // Numerator (* Jean-Fran├žois Alcover, May 18 2011 *)

LinearRecurrence[{0, 2, 0, 1}, {1, 0, 1, 2}, 40] (* and *) t = {1, 2}; Do[AppendTo[t, t[[-2]] + t[[-1]]]; AppendTo[t, t[[-3]] + t[[-1]]], {n, 30}]; t (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)

a0 := LinearRecurrence[{2, 1}, {1, 1}, 20]; (*     A001333 *)

a1 := LinearRecurrence[{2, 1}, {0, 2}, 20]; (* 2 * A000129 *)

Flatten[MapIndexed[{a0[[#]], a1[[#]]} &, Range[20]]] (* Gerry Martens, Jun 09 2015 *)

PROG

(PARI) x='x+O('x^50); Vec((1 - x^2 + 2*x^3)/(1 - 2*x^2 - x^4)) \\ G. C. Greubel, Oct 20 2017

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.

Sequence in context: A281839 A136570 A082766 * A082958 A218495 A166012

Adjacent sequences:  A119013 A119014 A119015 * A119017 A119018 A119019

KEYWORD

easy,frac,nonn

AUTHOR

Joshua Zucker, May 08 2006

STATUS

approved

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Last modified April 18 05:44 EDT 2021. Contains 343072 sequences. (Running on oeis4.)