|
| |
|
|
A007306
|
|
Denominators of Farey tree fractions (i.e. the Stern-Brocot subtree in the range [0,1]).
(Formerly M0437)
|
|
35
| |
|
|
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; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| 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
Apparently (except for the first term) the number of odd entries in the alternated diagonals of Pascal's triangle at 45 degrees slope. [Javier Torres (adaycalledzero(AT)hotmail.com), Jul 26 2009]
The Kn3 and Kn4 triangle sums, see A180662 for their definitions, of Sierpinski’s triangle A047999 equal A007306(n+1). [From Johannes W. Meijer, Jun 5 2011]
|
|
|
REFERENCES
| 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).
|
|
|
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
Alois P. Heinz, Table of n, a(n) for n = 0..10000
|
|
|
FORMULA
| For n > 0, a(n) = A002487(n-1) + A002487(n) = A002487(2*n-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, binomial(2*n-k, k) mod 2 ); a(n)=0^n+sum(k=0..n-1, binomial(2(n-1)-k, k) mod 2 ); - Paul Barry (pbarry(AT)wit.ie), Dec 11 2004
a(n)=sum(k=0..n, C(n+k,2*k) mod 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
| A007306 := proc(n): if n=0 then 1 else A002487(2*n-1) fi: end: A002487 := proc(m) option remember: local a, b, n; a := 1; b := 0; n := m; while n>0 do if type(n, odd) then b := a + b else a := a + b end if; n := floor(n/2); end do; b; end proc: seq(A007306(n), n=0..77); [From Johannes W. Meijer, Jun 5 2011]
|
|
|
MATHEMATICA
| a[0] = 1; a[n_] := Sum[ Mod[ Binomial[n+k-1, 2k] , 2], {k, 0, n}]; Table[a[n], {n, 0, 77}] (* From Jean-François Alcover, Dec 16 2011, after Paul Barry *)
|
|
|
PROG
| (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
Sequence in context: A026354 A179840 A078338 * A196155 A140858 A075458
Adjacent sequences: A007303 A007304 A007305 * A007307 A007308 A007309
|
|
|
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
Incorrect Maple program replaced by a correct one by Johannes W. Meijer, Jun 5 2011
|
| |
|
|
