0,5

Denominators of Farey fraction approximations to sqrt(2). The fractions are 1/0, 0/1, 1/1, 2/1, 3/2, 4/3, 7/5, 10/7, 17/12, .... See A082766(n+2) or A119016 for the numerators. "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. A097545/A097546 gives the similar sequence for pi. A119014/A119015 gives the similar sequence for e. - Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), May 09 2006

The principal and intermediate convergents to 2^(1/2) begin with 1/1, 3/2 4/3, 7/5, 10/7; essentially, numerators=A143607, denominators=A002965. - Clark Kimberling (ck6(AT)evansville.edu), Aug 27 2008

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

C. Brezinski, History of Continued Fractions and Pade' Approximants. Springer-Verlag, Berlin, 1991, p. 24.

H. S. M. Coxeter, The role of intermediate convergents in Tait's explanation for phyllotaxis, J. Algebra 20 (1972), 167-175.

Clark Kimberling, "Best lower and upper approximates to irrational numbers," Elemente der Mathematik, 52 (1997) 122-126.

Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.

K. Williams, The sacred cult revisited: the pavement of the baptistery of San Giovanni, Florence, Math. Intellig., 16 (No. 2, 1994), 18-24.

T. D. Noe, Table of n, a(n) for n=0..500

Pierre Lamothe, En marge du probleme des cercles tangents

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Dave Rusin, Farey fractions on sci.math

a(n) = 2*a(n-2)+a(n-4) if n>3; a(0)=0, a(1)=a(2)=a(3)=1.

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

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

a(0)=0, a(1)=1, a(n)=a(n-1)+a(2*[(n-2)/2]). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jan 31 2006

A002965 := proc(n) option remember; if n <= 0 then 0; elif n <= 3 then 1; else 2*A002965(n-2)+A002965(n-4); fi; end;

A002965:=-(1+2*z+z**2+z**3)/(-1+2*z**2+z**4); [Conjectured by S. Plouffe in his 1992 dissertation. Gives sequence except for two leading terms.]

(PARI) a(n)=if(n<4, n>0, 2*a(n-2)+a(n-4))

A000129(n) = A002965(2n), A001333(n) = A002965(2n+1).

Adjacent sequences: A002962 A002963 A002964 this_sequence A002966 A002967 A002968

Sequence in context: A123569 A048816 A080528 this_sequence A091696 A048808 A013983

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

Thanks to Michael Somos for several comments which improved this entry.