A006288 -id:A006288 - cp's OEIS frontend

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

Clark Kimberling, Dec 07 2008

nonn

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

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

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

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.

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

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

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 *)

{a(n) = if( n<-1 || n>0, 3*a(n\2) - (-1)^(n%2), 0)}; /* Michael Somos, Dec 22 2018 */

a(n) = fromdigits(apply(b->if(b,1,-1),binary(n)), 3); \\ Kevin Ryde, Feb 06 2022

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

A344892 Loxton-van der Poorten sequence: base-4 representation contains only -1, 0, +1, converted to ordinary base-4 digits 0,1,2,3.

Original entry on oeis.org

0, 1, 3, 10, 11, 23, 30, 31, 33, 100, 101, 103, 110, 111, 223, 230, 231, 233, 300, 301, 303, 310, 311, 323, 330, 331, 333, 1000, 1001, 1003, 1010, 1011, 1023, 1030, 1031, 1033, 1100, 1101, 1103, 1110, 1111, 2223, 2230, 2231, 2233, 2300, 2301, 2303, 2310, 2311

Offset: 0

Kevin Ryde, Jun 01 2021

Loxton and van der Poorten's morphism (see A344893), or the way -1 digits cause borrows, shows that this sequence is base 4 digit strings with no digit pair 12, 13, 20, or 21, and least significant digit not 2.

The least significant digit can be any of 0,1,3, then each successive higher digit has three choices: 0,1,3 above a 0 or 1, or 0,2,3 above a 2 or 3. This allows a(n) to be calculated by mapping from the ternary digits of n to these choices, from least to most significant digit.

Kevin Ryde, Table of n, a(n) for n = 0..6561

Cf. A006288 (decimal), A344893 (morphism), A007090 (base 4).

a(n) = my(v=digits(n,3),prev=0); forstep(i=#v,1,-1, prev=(v[i]+=(v[i]>(prev<2)))); fromdigits(v);

a(n) = A007090(A006288(n)).

A344893 Fixed point of the morphism 1->1321, 2->0021, 3->1300, 0->0000 starting from 1.

Original entry on oeis.org

1, 3, 2, 1, 1, 3, 0, 0, 0, 0, 2, 1, 1, 3, 2, 1, 1, 3, 2, 1, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 3, 2, 1, 1, 3, 2, 1, 1, 3, 0, 0, 0, 0, 2, 1, 1, 3, 2, 1, 1, 3, 2, 1, 1, 3, 0, 0, 0, 0, 2, 1, 1, 3, 2, 1, 1, 3, 2, 1, 1, 3, 0

Offset: 0

Kevin Ryde, Jun 01 2021

Loxton and van der Poorten give this morphism as a way to identify those n which can be represented in base 4 using only digits -1,0,+1 (A006288): n is a term of A006288 iff a(n) = 1 or 3.

Kevin Ryde, Table of n, a(n) for n = 0..8192
John Loxton and Alf van der Poorten, Arithmetic Properties of Automata: Regular Sequences, Journal für die Reine und Angewandte Mathematik, volume 392, 1988, pages 57-69. Also second author's copy. Section 1 example beta_n = a(n).
Index entries for 2-automatic sequences.
Index entries for sequences that are fixed points of mappings

Cf. A006288, A344892, A007090 (base 4).

Nest[Flatten[ReplaceAll[#,{0->{0,0,0,0},1->{1,3,2,1},2->{0,0,2,1},3->{1,3,0,0}}]]&,{1},4] (* Paolo Xausa, Nov 09 2023 *)

my(table=[9,8,9,0,0,8,6,2,4]); a(n) = my(s=2); if(n, forstep(i=bitor(logint(n,2),1),0,-1, (s=table[s-bittest(n,i)])||break)); s>>1;

a(n) = 0 if n in base 4 has a digit pair 12, 13, 20, or 21; otherwise a(n) = 1,3,2,1 according as n == 0,1,2,3 (mod 4).

A083905 G.f.: 1/(1-x) * sum(k>=0, (-1)^k*x^2^(k+1)/(1+x^2^k)).

Original entry on oeis.org

0, 1, 0, 0, -1, 1, 0, 1, 0, 2, 1, 0, -1, 1, 0, 0, -1, 1, 0, -1, -2, 0, -1, 1, 0, 2, 1, 0, -1, 1, 0, 1, 0, 2, 1, 0, -1, 1, 0, 2, 1, 3, 2, 1, 0, 2, 1, 0, -1, 1, 0, -1, -2, 0, -1, 1, 0, 2, 1, 0, -1, 1, 0, 0, -1, 1, 0, -1, -2, 0, -1, 1, 0, 2, 1, 0, -1, 1, 0, -1, -2, 0, -1, -2, -3, -1

Offset: 1

Ralf Stephan, Jun 18 2003

For all n, a(3*A006288(n)) = 0 as proved in Russian forum dxdy.ru - see link.

Discussion in Russian forum dxdy.ru
R. Stephan, Some divide-and-conquer sequences ...
R. Stephan, Table of generating functions
R. Stephan, Divide-and-conquer generating functions. I. Elementary sequences

Cf. A030300, A065359, A023416, A006288.

for(n=1, 100, l=ceil(log(n)/log(2)); t=polcoeff(1/(1-x)*sum(k=0, l, (-1)^k*(x^2^(k+1))/(1+x^2^k)) + O(x^(n+1)), n); print1(t", "))

a(n) = sum(i=0,logint(n,2)-1, if(!bittest(n,i),(-1)^i)); \\ Kevin Ryde, May 24 2021

a(1)=0, a(2n) = -a(n)+1, a(2n+1) = -a(n).

a(n) = A030300(n) - A065359(n).

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

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A344892 Loxton-van der Poorten sequence: base-4 representation contains only -1, 0, +1, converted to ordinary base-4 digits 0,1,2,3.

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A344893 Fixed point of the morphism 1->1321, 2->0021, 3->1300, 0->0000 starting from 1.

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A083905 G.f.: 1/(1-x) * sum(k>=0, (-1)^k*x^2^(k+1)/(1+x^2^k)).

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