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

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.

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

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).

Sequence in context: A123569 A048816 A080528 this_sequence A091696 A048808 A013983

Adjacent sequences: A002962 A002963 A002964 this_sequence A002966 A002967 A002968

nonn,easy,nice

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

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