|
| |
|
|
A050292
|
|
Maximal cardinality of a double-free subset of {1, 2, ..., n}.
|
|
20
|
|
|
|
0, 1, 1, 2, 3, 4, 4, 5, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 15, 16, 16, 17, 17, 18, 19, 20, 20, 21, 21, 22, 22, 23, 24, 25, 25, 26, 26, 27, 27, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 36, 36, 37, 37, 38, 38, 39, 40, 41, 41, 42, 43, 44, 44, 45, 46, 47, 47, 48, 48, 49, 49, 50, 51, 52, 52, 53, 54
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,4
|
|
|
COMMENTS
|
Maximal size of a subset S of {1, 2, ..., n} with the property that if x is in S then 2x is not.
Least k such that a(k)=n is equal to A003159(n).
To construct the sequence : let [a, b, c, a, a, a, b, c, a, b, c, ...] the fixed point of the morphism a -> abc, b ->a, c -> a, starting from a(1) = a, then write the indices of a, b, c that of a being written twice; see A092606 . - Philippe Deléham, Apr 13 2004
Number of integers from {1,...,n} for which the subtraction of 1 changes the parity of the number of 1's in their binary expansion. [From Vladimir Shevelev, Apr 15 2010]
Number of integers from {1,...,n} the factorization of which over different terms of A050376 does not contain 2. [From Vladimir Shevelev, Apr 16 2010]
a(0)=0 by convention.
a(n) modulo 2 is the Prouhet-Thue-Morse sequence A010060. - From Philippe Deléham, Oct 19 2011.
n appears A026465(n+1)times. From Philippe Deléham, Oct 19 2011.
|
|
|
REFERENCES
|
S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.26.
Wang, E. T. H. ``On Double-Free Sets of Integers.'' Ars Combin. 28, 97-100, 1989.
|
|
|
LINKS
|
_Reinhard Zumkeller_, Table of n, a(n) for n = 0..10000
S. R. Finch, Triple-Free Sets of Integers
R. Stephan, Some divide-and-conquer sequences ...
R. Stephan, Table of generating functions
Eric Weisstein's World of Mathematics, Double-Free Set.
|
|
|
FORMULA
|
Partial sums of A035263. Close to (2/3)*n.
a(1)=1, a(n)=n-a(floor(n/2)); a(n)=(2/3)*n+(1/3)*A065359(n); more generally, for m>=0, a(2^m*n)-2^m*a(n)=A001045(m)*A065359(n) where A001045(m)={2^m-(-1)^m}/3 is the Jacobsthal sequence; a(A039004(n))=(2/3)*A039004(n); a(2*A039004(n))=2*a(A039004(n)); a(A003159(n))=n; a(A003159(n)-1)=n-1; a(n)(mod 2)=A010060(n) the Thue-Morse sequence; a(n+1)-a(n)=A035263(n+1); a(n+2)-a(n) = abs(A029884(n)). - Benoit Cloitre, Nov 24, 2002
Series expansion: (1/(x*(x-1))) * Sum(i=0, infinity, (-1)^i*x^(2^i)/(x^(2^i)-1) ). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 17 2003
a(n)=sum(k=>0, (-1)^k*floor(n/2^k)) - Benoit Cloitre, Jun 03 2003
a(A091785(n)) = 2n; a(A091855(n)) = 2n-1 . - Philippe Deléham, Mar 26 2004
a(2^n)=(2^(n+1)+(-1)^n)/3 [From Vladimir Shevelev, Apr 15 2010]
If n=Sum{i>=0}b_i*2^i is the binary expansion of n, then a(n)=2n/3+(1/3)Sum{i>=0}b_i*(-1)^i. Thus a(n)=2n/3+O(log(n)) [From Vladimir Shevelev, Apr 15 2010]
Moreover, the equation a(3m)=2m has infinitely many solutions, e.g., a(3*2^k)=2*2^k; on the other hand, a((4^k-1)/3)=(2*(4^k-1))/9+k/3, i.e. limsup|a(n)-2n/3|=infinity. [From Vladimir Shevelev, Feb 23 2011]
a(n)=Sum_k>=0 {A030308(n,k)*A001045(k+1)}. - From Philippe Deléham, Oct 19 2011.
From Peter Bala, Feb 02 2013: (Start)
Product {n >= 1} (1 + x^((2^n - (-1)^n)/3 )) = (1 + x)^2(1 + x^3)(1 + x^5)(1 + x^11)(1 + x^21)... = 1 + sum {n >= 1} x^a(n) = 1 + 2x + x^2 + x^3 + 2x^4 + 2x^5 + .... Hence this sequence lists the numbers representable as a sum of distinct Jacobsthal numbers A001045 = [1, 1', 3, 5, 11, 21, ...], where we distinguish between the two occurrences of 1 by writing them as 1 and 1'. For example, 9 occurs twice in the present sequence because 9 = 5 + 3 + 1 and 9 = 5 + 3 + 1'. Cf. A197911 and A080277. See also A120385.
(End)
|
|
|
EXAMPLE
|
Examples for n = 1 through 8: {1}, {1}, {1,3}, {1,3,4}, {1,3,4,5}, {1,3,4,5}, {1,3,4,5,7}, {1,3,4,5,7}.
Since binary expansion of 5 is 101, then Sum{i>=0}b_i*(-1)^i=2. Therefore a(5)=10/3+2/3=4 [From Vladimir Shevelev, Apr 15 2010]
|
|
|
MATHEMATICA
|
a[n_] := a[n] = If[n < 2, 1, n - a[Floor[n/2]]]; Table[ a[n], {n, 1, 75}]
|
|
|
PROG
|
(PARI) a(n)=if(n<2, 1, n-a(floor(n/2)))
(Haskell)
a050292 n = a050292_list !! (n-1)
a050292_list = scanl (+) 0 a035263_list
-- Reinhard Zumkeller, Jan 21 2013
|
|
|
CROSSREFS
|
Cf. A001045, A050291-A050296, A050321, A035263.
Cf. A121701, A030308. A080277, A120385, A197911.
Sequence in context: A047784 A047742 A203967 * A181627 A071521 A225553
Adjacent sequences: A050289 A050290 A050291 * A050293 A050294 A050295
|
|
|
KEYWORD
|
nonn,nice,easy
|
|
|
AUTHOR
|
Eric W. Weisstein
|
|
|
EXTENSIONS
|
Extended with formula by Christian G. Bower, Sep 15 1999.
Corrected and extended by Reinhard Zumkeller, Aug 16 2006
Extended with formula by Philippe Deléham, Oct 19 2011.
|
|
|
STATUS
|
approved
|
| |
|
|
