0,5

Contribution from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 22 2008: (Start)

For n>1: a(n+2) = if A025480(n-1)<>0 and A025480(n)<>0 then a(A025480(n-1)+2)+a(A025480(n)+2) else if A025480(n)=0 then a(A025480(n-1)+2)+1 else 0+a(A025480(n-1)+2);

a(A054429(n)+2) = A047679(n) and a(n+2) = A047679(A054429(n));

A153036(n) = floor(a(n+2)/A047679(n)). (End)

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

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 117.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 23.

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.

W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 154.

G. Melancon, Lyndon factorization of sturmian words, Discr. Math., 210 (2000), 137-149.

I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 141.

T. D. Noe, Table of n, a(n) for n=0..4096

Index entries for sequences related to Stern's sequences

A. Bogomolny, Stern-Brocot Tree

A. Bogomolny, Inspiration for Maple code

N. J. A. Sloane, Stern-Brocot or Farey Tree

G. A. Jones, The Farey graph

a(n) = SternBrocotTreeNum(n-1) # n starting from 2 gives the sequence from 1, 1, 2, 1, 2, 3, 3, 1, 2, 3, 3, 4, 5, 5, 4, 1, ...

[ 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;...

SternBrocotTreeNum := proc(n) option remember; local msb, r; if(n < 2) then RETURN(n); fi; msb := floor_log_2(n); r := n - (2^msb); if(floor_log_2(r) = (msb-1)) then RETURN(SternBrocotTreeNum(r) + SternBrocotTreeNum(((3*(2^(msb-1)))-r)-1)); else RETURN(SternBrocotTreeNum((2^(msb-1))+r)); fi; end;

Contribution from Peter Luschny (peter(AT)luschny.de), Apr 27 2009: (Start)

sbt[n_] := Module[{R, L, Y}, R={{1, 0}, {1, 1}}; L={{1, 1}, {0, 1}}; Y={{1, 0}, {0, 1}}; w[b_] := Fold[ #1.If[ #2 == 0, L, R] &, Y, b]; u[a_] := {a[[2, 1]]+a[[2, 2]], a[[1, 1]]+a[[1, 2]]}; Map[u, Map[w, Tuples[{0, 1}, n]]]]

A007305(n) = Flatten[Append[{0, 1}, Table[Map[First, sbt[i]], {i, 0, 5}]]]

A047679(n) = Flatten[Table[Map[Last, sbt[i]], {i, 0, 5}]] (End)

Cf. A007306, A006842, A006843, A047679, A054424, A057114.

A152975. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 22 2008]

Adjacent sequences: A007302 A007303 A007304 this_sequence A007306 A007307 A007308

Sequence in context: A035531 A118977 A071766 this_sequence A112531 A100002 A057041

nonn,frac,tabf,nice

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

Maple code from Antti Karttunen, Mar 19 2000