A147991 Sequence S such that 1 is in S and if x is in S, then 3x-1 and 3x+1 are in S.

1, 2, 4, 5, 7, 11, 13, 14, 16, 20, 22, 32, 34, 38, 40, 41, 43, 47, 49, 59, 61, 65, 67, 95, 97, 101, 103, 113, 115, 119, 121, 122, 124, 128, 130, 140, 142, 146, 148, 176, 178, 182, 184, 194, 196, 200, 202, 284, 286, 290, 292, 302, 304, 308, 310, 338, 340, 344, 346

Offset: 1

Author

Clark Kimberling, Dec 07 2008

Keywords

• nonn

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

Examples

			0th generation: 1;
1st generation: 2 4;
2nd generation: 5 7 11 13.
		

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

Crossrefs

Cf. A168542, A191108, A306556.

Cf. A006288, A351243 (non-quotients).

See also the related sequences listed in A191106.

One half of each position > 0 where A307744 sets or equals a record.

Cf. A030300.

Column k=3 of A360099.

Programs

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

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 *)

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

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

For n > 0, A307744(2*a(2n)) = A307744(2*a(2n+1)) = A307744(2*a(n)) + 1. - Peter Munn, Jan 31 2022

a(n) mod 2 = A030300(n). - Alois P. Heinz, Jan 29 2023