Search: id:A007306
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%I A007306 M0437
%S A007306 1,1,2,3,3,4,5,5,4,5,7,8,7,7,8,7,5,6,9,11,10,11,13,12,9,9,12,13,11,10,
%T A007306 11,9,6,7,11,14,13,15,18,17,13,14,19,21,18,17,19,16,11,11,16,19,17,18,
%U A007306 21,19,14,13,17,18,15,13,14,11,7,8,13,17,16,19,23,22,17,19,26,29,25,24
%N A007306 Denominators of Farey tree fractions (i.e. the Stern-Brocot subtree in 
               the range [0,1]).
%C A007306 Also number of odd entries in n-th row of triangle of Stirling numbers 
               of the second kind (A008277). - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Feb 28 2004
%C A007306 Contribution from Javier Torres (adaycalledzero(AT)hotmail.com), Jul 
               26 2009: (Start)
%C A007306 It appears that are also the odd entries in alternated diagonals in Pascal's 
               triangle at 45 degrees slope
%C A007306 (End)
%D A007306 P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 
               1968, vol. 2, p. 61.
%D A007306 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 
               256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see 
               vol. 1, p. 158.
%D A007306 J. C. Lagarias, Number Theory and Dynamical Systems, pp. 35-72 of S. 
               A. Burr, ed., The Unreasonable Effectiveness of Number Theory, Proc. 
               Sympos. Appl. Math., 46 (1992). Amer. Math. Soc.
%D A007306 G. Melancon, Lyndon factorization of sturmian words, Discr. Math., 210 
               (2000), 137-149.
%D A007306 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A007306 A. Bogomolny, <a href="http://www.cut-the-knot.org/blue/Stern.shtml">
               Stern-Brocot tree</a>
%H A007306 N. J. A. Sloane, <a href="stern_brocot.html">Stern-Brocot or Farey Tree</
               a>
%H A007306 Javier Torres Suarez, <a href="http://www.youtube.com/watch?v=__Re3zKM9n8&feature=related">
               Number theory - geometric connection (part 2)</a> (YouTube video 
               that mentions this sequence - link sent by Pacha Nambi, Aug 26 2009)
%H A007306 <a href="Sindx_St.html#Stern">Index entries for sequences related to 
               Stern's sequences</a>
%F A007306 For n > 0, a(n) = A002487(n-1) + A002487(n) = A002487(2n-1).
%F A007306 a(0)=1; for n>=1 a(n)=sum(k=0, n-1, C(n-1+k, n-1-k) mod 2 ) - Benoit 
               Cloitre (benoit7848c(AT)orange.fr), Jun 20 2003
%F A007306 a(n+1)=sum{k=0..n, mod(binomial(2n-k, k), 2)}; a(n)=0^n+sum{k=0..n-1, 
               mod(binomial(2(n-1)-k, k), 2)}; - Paul Barry (pbarry(AT)wit.ie), 
               Dec 11 2004
%F A007306 a(n)=sum{k=0..n, mod(C(n+k,2k),2)}; - Paul Barry (pbarry(AT)wit.ie), 
               Jun 12 2006
%e A007306 [ 0/1; 1/1; ] 1/2; 1/3, 2/3; 1/4, 2/5, 3/5, 3/4; 1/5, 2/7, 3/8, 3/7, 
               4/7, 5/8, 5/7, 4/5;...
%p A007306 SB01Den := proc(n) option remember; local r; if(n <= 1) then RETURN(n+1); 
               fi; r := n - 2^floor_log_2(n); if(0 = (floor((1+r)/2) mod 2)) then 
               RETURN(2*SB01Den(floor(n/2)) - SB01Den(floor(n/4))); else RETURN(SB01Den(floor(n/
               2)) + SB01Den(floor(n/4))); fi; end;
%p A007306 [seq(SB01Den(n), n=0..64)]; # starts as [1,2,3,3,4,5,5,...]
%o A007306 (PARI) a(n)=if(n<1, n==0, n--; sum(k=0,n, binomial(n+k,n-k)%2))
%o A007306 (PARI) a(n)=local(m); if(n<2, n>=0, m=2^length(binary(n-1)); a(n-m/2)+a(m-n+1)) 
               /* Michael Somos May 30 2005 */
%Y A007306 Cf. A007305, A006842, A006843, A047679, A054424, A065674-A065675, A065810
%Y A007306 Sequence in context: A115728 A026354 A078338 this_sequence A140858 A075458 
               A083036
%Y A007306 Adjacent sequences: A007303 A007304 A007305 this_sequence A007307 A007308 
               A007309
%K A007306 nonn,frac,tabf,nice
%O A007306 0,3
%A A007306 N. J. A. Sloane (njas(AT)research.att.com).
%E A007306 Formula fixed and extended by Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), 
               Jul 07 2009

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