A055247 -id:A055247 - cp's OEIS frontend

A055246 At step number k >= 1 the 2^(k-1) open intervals that are erased from [0,1] in the Cantor middle-third set construction are I(k,n) = (a(n)/3^k, (1+a(n))/3^k), n=1..2^(k-1).

1, 7, 19, 25, 55, 61, 73, 79, 163, 169, 181, 187, 217, 223, 235, 241, 487, 493, 505, 511, 541, 547, 559, 565, 649, 655, 667, 673, 703, 709, 721, 727, 1459, 1465, 1477, 1483, 1513, 1519, 1531, 1537, 1621, 1627, 1639, 1645, 1675, 1681, 1693, 1699

Offset: 1

Wolfdieter Lang, May 23 2000

Related to A005836. Gives boundaries of open intervals that have to be erased in the Cantor middle-third set construction.
Let g(n) = Sum_{i=0..n} (i*binomial(n+i,i)^3*binomial(n,i)^2) = A112035(n). Let b = {m>0 : g(m) != 0 (mod 3)}. Then b(n) = a(n). - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 08 2004
Conjecture: Similarly to A191107, this increasing sequence is generated by the rules: a(1) = 1, and if x is in the sequence, then 3*x-2 and 3*x+4 are also in the sequence. - L. Edson Jeffery, Nov 17 2015

			k=1: (1/3, 2/3);
k=2: (1/9, 2/9), (7/9, 8/9);
k=3: (1/27, 2/27), (7/27, 8/27), (19/27, 20/27), (25/27, 26/27); ...

Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Index entries for 3-automatic sequences.

Cf. A003278, A005836, A005823, A055247, A112035.

Cf. A191107.

(* (Conjectured) Choose rows large enough to guarantee that all terms < max are generated. *)
rows = 1000; max = 10^4; a[1] = {1}; i = 1; Do[a[n_] = {}; Do[If[1 < 3*a[n - 1][[k]] - 2 < max, AppendTo[a[n], 3*a[n - 1][[k]] - 2], Break]; If[3*a[n - 1][[k]] + 4 < max, AppendTo[a[n], 3*a[n - 1][[k]] + 4], Break], {k, Length[a[n - 1]]}]; If[a[n] == {}, Break, i++], {n, 2, 1000}]; a055246 = Take[Flatten[Table[a[n], {n, i}]], 48] (* L. Edson Jeffery, Nov 17 2015 *)
Join[{1}, 1 + 6 Accumulate[Table[(3^IntegerExponent[n, 2] + 1)/2, {n, 60}]]] (* Vincenzo Librandi, Nov 26 2015 *)

g(n)=sum(i=0,n,i*binomial(n+i,i)^3*binomial(n,i)^2);
for (i=1,2000,if(Mod(g(i),3)<>0,print1(i,",")))

a(n) = fromdigits(binary(n-1),3)*6 + 1; \\ Kevin Ryde, Apr 23 2021

def A055246(n): return int(bin(n-1)[2:],3)*6|1 # Chai Wah Wu, Jun 26 2025

a(n) = 1+6*A005836(n), n >= 1.
a(n) = 1+3*A005823(n), n >= 1.
a(n+1) = A074938(n) + A074939(n); A074938: odd numbers in A005836, A074939: even numbers in A005836. - Philippe Deléham, Jul 10 2005
Conjecture: a(n) = 2*A191107(n) - 1 = 6*A003278(n) - 5 = (a((2*n-1)*2^(k-1))+2)/3^k, k>0. - L. Edson Jeffery, Nov 25 2015

Edited by N. J. A. Sloane, Nov 20 2015: used first comment to give more precise definition, and edited a comment at the suggestion of L. Edson Jeffery.

A191108 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x-2 and 3x+2 are in a.

Original entry on oeis.org

1, 5, 13, 17, 37, 41, 49, 53, 109, 113, 121, 125, 145, 149, 157, 161, 325, 329, 337, 341, 361, 365, 373, 377, 433, 437, 445, 449, 469, 473, 481, 485, 973, 977, 985, 989, 1009, 1013, 1021, 1025, 1081, 1085, 1093, 1097, 1117, 1121, 1129, 1133, 1297, 1301, 1309, 1313, 1333, 1337, 1345, 1349, 1405, 1409, 1417, 1421, 1441, 1445

Offset: 1

Clark Kimberling, May 26 2011

See discussions at A190803, A191106.  The sequence a=A191108 has closure properties:  the positive integers in (2+A191108)/3 comprise A191108, as do those in (-2+A191108)/3.
From Peter Munn, May 13 2019: (Start)
The closure of {1} in the positive integers under reflection about 3^k, k >= 1.
Asymptotic density is 0.
Consider a Sierpinski arrowhead curve formed of edges numbered consecutively from 0 at its axis of symmetry. The m-th edge is contained in the boundary of the plane sector occupied by the arrowhead if and only if m or -m is in this sequence.
For k >= 0, a(2^k) = 2*3^k - 1 and {a(i)/(2*3^k) | 1 <= i <= 2^k} is the set of center points of surviving intervals at the k-th step of generating the Cantor set, and therefore the set of center points of deleted middle-third intervals at the (k+1)-th step.
Define t: Z -> P(R) so that t(n) is the translated Cantor ternary set spanning [(n-1)/2, (n+1)/2], and let T be the union of t(a(n)) for all n. T = T * 3 = T / 3 is the closure of the Cantor ternary set under multiplication by 3.
(End)

Ivan Neretin, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Cantor Set
Wikipedia, Sierpiński arrowhead curve

Cf. A005823, A005836, A055247, A147991, A190803, A191106, A306556.

h = 3; i = -2; j = 3; k = 2; f = 1; g = 7;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* A191108 *)
b = (a + 2)/3; c = (a - 2)/3; r = Range[1, 900];
d = Intersection[b, r] (* A191108 closure property  *)
e = Intersection[c, r] (* A191108 closure property  *)

a(n) = fromdigits(binary(n-1),3)<<2 + 1; \\ Kevin Ryde, Aug 05 2022

From Peter Munn, May 25 2019: (Start)
a(n) = (A055247(2n-1) + A055247(2n)) / 3.
a(n) = A306556(2n)*2 - 1 = A306556(2n-1) + A306556(2n).
a(n) = 2*A005823(n) + 1 = 4*A005836(n) + 1 = 2*A191106(n) - 1.
a(2^k+i) = 2*A147991(2^k+i-1) + 3^(k+1) for k >= 0, 1 <= i <= 2^k.
(End)

A306556 Integers that appear as (unreduced) numerators of segment endpoints when a ternary Cantor set is created.

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 8, 9, 18, 19, 20, 21, 24, 25, 26, 27, 54, 55, 56, 57, 60, 61, 62, 63, 72, 73, 74, 75, 78, 79, 80, 81, 162, 163, 164, 165, 168, 169, 170, 171, 180, 181, 182, 183, 186, 187, 188, 189, 216, 217, 218, 219, 222, 223, 224, 225, 234, 235, 236, 237, 240, 241, 242, 243

Offset: 1

Dan Dima, Feb 23 2019

Nonnegative integers whose ternary representation contains only digits 0 and 2 except for at most a single digit 1 that is followed only by 0's.
Nonnegative integers that can be written in base 3 using only 0's and 2's, allowing the use of the "decimal" point (.) and replacing ....10..0(.) by ....02..2(.)2222...
Note that fractions are not reduced.
List of integers in the closure of the ternary Cantor set under multiplication by 3. The closure is the union of the translated ternary Cantor sets spanning [a(1), a(2)], [a(3), a(4)], [a(5), a(6)], ... . - Peter Munn, Jul 09 2019

			On 1st step we have [0,1/3] U [2/3,3/3] so we get a(1)=0, a(2)=1, a(3)=2, a(4)=3.
On 2nd step we have [0,1/9] U [2/9,3/9] U [6/9,7/9] U [8/9,9/9] so we get in addition a(5)=6, a(6)=7, a(7)=8, a(8)=9.

Georg Cantor, Über unendliche, lineare Punktmannigfaltigkeiten V" [On infinite, linear point-manifolds (sets), Part 5]. Mathematische Annalen (in German). (1883) 21: 545-591.
Paul du Bois-Reymond, Der Beweis des Fundamentalsatzes der Integralrechnung, Mathematische Annalen (in German), (1880), 16, footnote on p. 128.
Eric Weisstein's World of Mathematics, Cantor Set
Wikipedia, Cantor set
Index entries for 3-automatic sequences.

Cf. A005823, A054591, A055247, A191106, A191108.

A306556(n) = {sm=0;while(n>1,ex=floor(log(n)/log(2));if(n-2^ex==0,sm=sm+3^(ex-1),sm=sm+2*3^(ex-1));n=n-2^ex);return(sm)}

a(n) = n--; fromdigits(binary(n>>1),3)*2 + (n%2); \\ Kevin Ryde, Apr 23 2021

a(1)=0, a(2)=1;
a(2^n) = 3^(n-1) for n >= 1;
a(2^n+k) = 2*3^(n-1) + a(k) for 1 <= k <= 2^n.
From Peter Munn, Jul 09 2019: (Start)
a(2n-1) = A005823(n) = A191106(n)-1.
a(2n)   = A191106(n) = A005823(n)+1.
a(2n-1) = (A055247(2n-1)-1)/3.
a(2n)   = (A055247(2n)  +1)/3.
a(2n-1) = (A191108(n)-1)/2.
a(2n)   = (A191108(n)+1)/2.
(End)

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A055246 At step number k >= 1 the 2^(k-1) open intervals that are erased from [0,1] in the Cantor middle-third set construction are I(k,n) = (a(n)/3^k, (1+a(n))/3^k), n=1..2^(k-1).

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A191108 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x-2 and 3x+2 are in a.

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A306556 Integers that appear as (unreduced) numerators of segment endpoints when a ternary Cantor set is created.

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