OFFSET
1,2
COMMENTS
Positive numbers that can be written in balanced ternary without a 0 trit. - J. Hufford, Jun 30 2015
Let S be the set of terms. Define c: Z -> P(R) so that c(m) is the translated Cantor ternary set spanning [m-0.5, m+0.5], and let C be the union of c(m) for all m in S U {0} U -S. C is the closure of the translated Cantor ternary set spanning [-0.5, 0.5] under multiplication by 3. - Peter Munn, Jan 31 2022
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
Gevorg Hmayakyan, Trig identity for a(n)
J. H. Loxton and A. J. van der Poorten, An Awful Problem About Integers in Base Four, Acta Arithmetica, volume 49, 1987, pages 193-203. In section 7, John Selfridge and Carole Lacampagne ask whether every k != 0 (mod 3) is the quotient of two terms of this sequence (cf. A351243 and A006288).
Eric Weisstein's World of Mathematics, Cantor Set
Eric Weisstein's World of Mathematics, Closure
FORMULA
a(n) = 3*a(n/2) - 1 if n>=2 is even, 3*a((n-1)/2) + 1 if n is odd, a(0)=0. - Robert Israel, May 05 2014
G.f. g(x) satisfies g(x) = 3*(x+1)*g(x^2) + x/(1+x). - Robert Israel, May 05 2014
Product_{j=0..n-1} cos(3^j) = 2^(-n+1)*Sum_{i=2^(n-1)..2^n-1} cos(a(i)). - Gevorg Hmayakyan, Jan 15 2017
Sum_{i=2^(n-1)..2^n-1} cos(a(i)/3^(n-1)*Pi/2) = 0. - Gevorg Hmayakyan, Jan 15 2017
a(n) = -a(-1-n) for all n in Z. - Michael Somos, Dec 22 2018
EXAMPLE
0th generation: 1;
1st generation: 2 4;
2nd generation: 5 7 11 13.
MAPLE
A147991:= proc(n) option remember; if n::even then 3*procname(n/2)-1 else 3*procname((n-1)/2)+1 fi end proc:
A147991(1):= 1:
[seq](A147991(i), i=1..1000); # Robert Israel, May 05 2014
MATHEMATICA
nn=346; s={1}; While[s1=Select[Union[s, 3*s-1, 3*s+1], # <= nn &]; s != s1, s=s1]; s
a[ n_] := If[ n < -1 || n > 0, 3 a[Quotient[n, 2]] - (-1)^Mod[n, 2], 0]; (* Michael Somos, Dec 22 2018 *)
PROG
(Haskell)
import Data.Set (singleton, insert, deleteFindMin)
a147991 n = a147991_list !! (n-1)
a147991_list = f $ singleton 1 where
f s = m : (f $ insert (3*m - 1) $ insert (3*m + 1) s')
where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Feb 21 2012, Jan 23 2011
(PARI) {a(n) = if( n<-1 || n>0, 3*a(n\2) - (-1)^(n%2), 0)}; /* Michael Somos, Dec 22 2018 */
(PARI) a(n) = fromdigits(apply(b->if(b, 1, -1), binary(n)), 3); \\ Kevin Ryde, Feb 06 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Dec 07 2008
STATUS
approved