%I A002965 M0671
%S A002965 0,1,1,1,2,3,5,7,12,17,29,41,70,99,169,239,408,577,985,1393,2378,3363,
%T A002965 5741,8119,13860,19601,33461,47321,80782,114243,195025,275807,470832,
%U A002965 665857,1136689,1607521,2744210,3880899,6625109,9369319,15994428
%N A002965 Interleave denominators (A000129) and numerators (A001333) of convergents 
               to sqrt(2).
%C A002965 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
%C A002965 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
%D A002965 C. Brezinski, History of Continued Fractions and Pade' Approximants. 
               Springer-Verlag, Berlin, 1991, p. 24.
%D A002965 H. S. M. Coxeter, The role of intermediate convergents in Tait's explanation 
               for phyllotaxis, J. Algebra 20 (1972), 167-175.
%D A002965 Clark Kimberling, "Best lower and upper approximates to irrational numbers,
               " Elemente der Mathematik, 52 (1997) 122-126.
%D A002965 Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, 
               New York, 1966.
%D A002965 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002965 K. Williams, The sacred cult revisited: the pavement of the baptistery 
               of San Giovanni, Florence, Math. Intellig., 16 (No. 2, 1994), 18-24.
%H A002965 T. D. Noe, <a href="b002965.txt">Table of n, a(n) for n=0..500</a>
%H A002965 Pierre Lamothe, <a href="http://www.aei.ca/~plamothe/tangents.htm">En 
               marge du probleme des cercles tangents</a>
%H A002965 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A002965 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A002965 Dave Rusin, <a href="http://www.math.niu.edu/~rusin/known-math/99/farey">
               Farey fractions on sci.math</a>
%F A002965 a(n) = 2*a(n-2)+a(n-4) if n>3; a(0)=0, a(1)=a(2)=a(3)=1.
%F A002965 a(2n)=a(2n-1)+a(2n-2), a(2n+1)=2a(2n)-a(2n-1).
%F A002965 G.f.: (x+x^2-x^3)/(1-2*x^2-x^4).
%F A002965 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
%p A002965 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;
%p A002965 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.]
%o A002965 (PARI) a(n)=if(n<4,n>0,2*a(n-2)+a(n-4))
%Y A002965 A000129(n) = A002965(2n), A001333(n) = A002965(2n+1).
%Y A002965 Sequence in context: A123569 A048816 A080528 this_sequence A091696 A048808 
               A013983
%Y A002965 Adjacent sequences: A002962 A002963 A002964 this_sequence A002966 A002967 
               A002968
%K A002965 nonn,easy,nice
%O A002965 0,5
%A A002965 N. J. A. Sloane (njas(AT)research.att.com).
%E A002965 Thanks to Michael Somos for several comments which improved this entry.