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Search: id:A007306
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| A007306 |
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Denominators of Farey tree fractions (i.e. the Stern-Brocot subtree in the range [0,1]).
(Formerly M0437)
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+0
31
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| 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, 11, 9, 6, 7, 11, 14, 13, 15, 18, 17, 13, 14, 19, 21, 18, 17, 19, 16, 11, 11, 16, 19, 17, 18, 21, 19, 14, 13, 17, 18, 15, 13, 14, 11, 7, 8, 13, 17, 16, 19, 23, 22, 17, 19, 26, 29, 25, 24
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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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
Contribution from Javier Torres (adaycalledzero(AT)hotmail.com), Jul 26 2009: (Start)
It appears that are also the odd entries in alternated diagonals in Pascal's triangle at 45 degrees slope
(End)
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REFERENCES
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P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 61.
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.
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.
G. Melancon, Lyndon factorization of sturmian words, Discr. Math., 210 (2000), 137-149.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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A. Bogomolny, Stern-Brocot tree
N. J. A. Sloane, Stern-Brocot or Farey Tree
Javier Torres Suarez, Number theory - geometric connection (part 2) (YouTube video that mentions this sequence - link sent by Pacha Nambi, Aug 26 2009)
Index entries for sequences related to Stern's sequences
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FORMULA
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For n > 0, a(n) = A002487(n-1) + A002487(n) = A002487(2n-1).
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
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
a(n)=sum{k=0..n, mod(C(n+k,2k),2)}; - Paul Barry (pbarry(AT)wit.ie), Jun 12 2006
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EXAMPLE
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[ 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;...
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MAPLE
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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;
[seq(SB01Den(n), n=0..64)]; # starts as [1, 2, 3, 3, 4, 5, 5, ...]
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PROGRAM
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(PARI) a(n)=if(n<1, n==0, n--; sum(k=0, n, binomial(n+k, n-k)%2))
(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 */
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CROSSREFS
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Cf. A007305, A006842, A006843, A047679, A054424, A065674-A065675, A065810
Sequence in context: A115728 A026354 A078338 this_sequence A140858 A075458 A083036
Adjacent sequences: A007303 A007304 A007305 this_sequence A007307 A007308 A007309
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KEYWORD
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nonn,frac,tabf,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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Formula fixed and extended by Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jul 07 2009
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