|
Search: id:A007306
|
|
|
| A007306 |
|
Denominators of Farey tree fractions (i.e. the Stern-Brocot subtree in the range [0,1]).
(Formerly M0437)
|
|
+0
31
|
|
| 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)
|
|
|
OFFSET
|
0,3
|
|
|
COMMENT
|
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)
|
|
REFERENCES
|
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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.
|
|
LINKS
|
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
|
|
FORMULA
|
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
|
|
EXAMPLE
|
[ 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;...
|
|
MAPLE
|
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, ...]
|
|
PROGRAM
|
(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 */
|
|
CROSSREFS
|
Cf. A007305, A006842, A006843, A047679, A054424, A065674-A065675, A065810
Adjacent sequences: A007303 A007304 A007305 this_sequence A007307 A007308 A007309
Sequence in context: A115728 A026354 A078338 this_sequence A140858 A075458 A083036
|
|
KEYWORD
|
nonn,frac,tabf,nice
|
|
AUTHOR
|
N. J. A. Sloane (njas(AT)research.att.com).
|
|
EXTENSIONS
|
Formula fixed and extended by Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jul 07 2009
|
|
|
Search completed in 0.003 seconds
|
