A061393 -id:A061393 - cp's OEIS frontend

A191106 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x-2 and 3x are in a.

1, 3, 7, 9, 19, 21, 25, 27, 55, 57, 61, 63, 73, 75, 79, 81, 163, 165, 169, 171, 181, 183, 187, 189, 217, 219, 223, 225, 235, 237, 241, 243, 487, 489, 493, 495, 505, 507, 511, 513, 541, 543, 547, 549, 559, 561, 565, 567, 649, 651, 655, 657, 667, 669, 673, 675, 703, 705, 709, 711, 721, 723, 727, 729, 1459, 1461, 1465, 1467, 1477

Offset: 1

Clark Kimberling, May 26 2011

Related sequences for various choices of i and k as defined in A190803:
  A003278:  (i,k) = (-2,-1)
  A191106:  (i,k) = (-2, 0)
  A191107:  (i,k) = (-2, 1)
  A191108:  (i,k) = (-2, 2)
  A153775:  (i,k) = (-1, 0)
  A147991:  (i,k) = (-1, 1)
  A191109:  (i,k) = (-1, 2)
  A005836:  (i,k) = ( 0, 1)
  A191110:  (i,k) = ( 0, 2)
  A132140:  (i,k) = ( 1, 2)
For a=A191106, we have closure properties: the integers in (2+a)/3 comprise a; the integers in a/3 comprise a.
For k >= 1, m = a(i), 1 <= i <= 2^k seems to be m such that m/(3^k+1) is in the Cantor set (except that m = 0 and m = 3^k+1 do not appear). For k >= 2,  m = (a(i)-1)/2, 1 <= i <= 2^k seems to be m such that m/((3^k-1)/2) is in the Cantor set. - Peter Munn, Jul 06 2019
Every even number is the sum of two (possibly equal) terms. More specifically: terms a(1) through a(2^n) = 3^n sum to even numbers 2 times 1 through 3^n. Every even number is infinitely often the difference of two terms. Since the sequence is equal to 2*A005836(n) + 1, these properties follow immediately from similar properties of A005836 for every number. - Aad Thoen, Feb 17 2022
if A_n=(a(1),a(2),...,a(2^n)), then A_(n+1)=(A_n,A_n+2*3^n), similar to A003278. - Arie Bos, Jul 26 2022

			1 -> 3 -> 7,9 -> 19,21,25,27 -> ...

Ivan Neretin, Table of n, a(n) for n = 1..10000
David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, 10 (2007) 1-13.
D. Jordan and R. Schayer Rational points on the Cantor middle thirds set, Penn State, REU 2003.
Eric Weisstein's World of Mathematics, Cantor Set

Cf. A005823, A005836, A054591, A088917 (characteristic function), A173934, A190803, A191108.
Partial sums of A061393.
Similar formula as A003278, A_(n+1)=(A_n,A_n+2*3^n).

h = 3; i = -2; j = 3; k = 0; f = 1; g = 9;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* A191106; regarding g, see note at A190803 *)
b = (a + 2)/3; c = a/3; r = Range[1, 900];
d = Intersection[b, r](* illustrates closure property *)
e = Intersection[c, r](* illustrates closure property *)
2 FromDigits[#, 3]&/@Tuples[{0, 1}, 7] + 1 (* Vincenzo Librandi, Jul 10 2019 *)

a(n) = 2*A005836(n) + 1. - Charles R Greathouse IV, Sep 06 2011
a(n) = A005823(n) + 1. - Vladimir Shevelev, Dec 17 2012
a(n) = (A191108(n) + 1)/2. - Peter Munn, Jul 09 2019

A061392 a(n) = a(floor(n/3)) + a(ceiling(n/3)) with a(0) = 0 and a(1) = 1.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 16, 16, 16, 16, 16, 16, 16

Offset: 0

Henry Bottomley, Apr 30 2001

Number of nonnegative integers < n having no 1 in their ternary representation. - Reinhard Zumkeller, Mar 23 2003; corrected by Henry Bottomley, Mar 24 2003

Rémy Sigrist, Table of n, a(n) for n = 0..6561
Sam Northshield, Sums across Pascal's triangle modulo 2, Congressus Numerantium, 200, pp. 35-52, 2010.
Wikipedia, Cantor function.

k appears A061393(k) times.
Cf. A007089, A062756, A081608, A081609, A081611.
Essentially the partial sums of A088917.

A061392 := proc(n) option remember; local a : if n <=1 then n else A061392(floor(n/3)) + A061392(ceil(n/3)) fi: end: seq(A061392(n), n=0..87); # Johannes W. Meijer, Jun 05 2011

a(n+1) + A081609(n) = n+1. - Reinhard Zumkeller, Mar 23 2003; corrected by Henry Bottomley, Mar 24 2003
From Johannes W. Meijer, Jun 05 2011: (Start)
a(3*n+1) = a(n+1) + a(n), a(3*n+2) = a(n+1) + a(n) and a(3*n+3) = 2*a(n+1), for n>=1, with a(0)=0, a(1)=1, a(2)=1 and a(3)=2. [Northshield]
G.f.: x*Product_{n>=0} (1 + x^(3^n) + 2*x^(2*3^n) + x^(3*3^n) + x^(4*3^n)). [Northshield] (End)
Apparently, for any n >= 0 and k such that n < 3^k, a(n) = 2^k * c(n / 3^k) where c is the Cantor function. - Rémy Sigrist, Jul 12 2019

A236313 Recurrence: a(2n) = 3a(n)-1, a(2n+1) = 1.

Original entry on oeis.org

1, 2, 1, 5, 1, 2, 1, 14, 1, 2, 1, 5, 1, 2, 1, 41, 1, 2, 1, 5, 1, 2, 1, 14, 1, 2, 1, 5, 1, 2, 1, 122, 1, 2, 1, 5, 1, 2, 1, 14, 1, 2, 1, 5, 1, 2, 1, 41, 1, 2, 1, 5, 1, 2, 1, 14, 1, 2, 1, 5, 1, 2, 1, 365, 1, 2, 1, 5, 1, 2, 1, 14, 1, 2, 1, 5, 1, 2, 1, 41, 1, 2, 1, 5, 1, 2, 1, 14, 1, 2, 1, 5, 1, 2, 1, 122, 1, 2, 1, 5

Offset: 1

Ralf Stephan, Jan 22 2014

First differences of Stanley sequence S(0,1) (A005836) and S(1,2) (A003278).
In the binary expansion of n, delete everything left of the rightmost 1 bit, then interpret as ternary, add one, and divide by 2.
A007051 is this sequence in strictly increasing order. - Max Barrentine, Sep 11 2015
Empirical: a(n) is the smallest natural number k such that no two adjacent subsequences t and u consisting of consecutive entries of (a(1), a(2), ..., a(n-1), k) are such that the sum of the entries of t is equal to the sum of the entries of u. For example, according to this definition, a(4) cannot be equal to 1, 2, 3, or 4. - John M. Campbell, Mar 20 2017

Cf. A003278, A005836, A006519, A007051, A007814, A061393, A093682.

[(1+3^Valuation(n,2))/2: n in [1..100]]; // Bruno Berselli, Jan 22 2014

t = {1}; Do[If[OddQ[n], AppendTo[t, 1], AppendTo[t, 3*t[[n/2]] - 1]], {n, 2, 100}]; t (* T. D. Noe, Apr 10 2014 *)
a[n_] := a[n] =If[OddQ@ n, 1, 3 a[n/2] - 1]; Array[a, 92] (* Robert G. Wilson v, Jul 31 2018 *)

PARI
```
a(n)=(1+3^valuation(n,2))/2
```

a(n) = (1/2)*(1 + 3^A007814(n)) = A007051(A007814(n)).
a(n) = (1/2)*A061393(n), for n>=1.
Multiplicative with a(2^e) = (1 + 3^e)/2, a(p^e) = 1 for odd prime p. - Andrew Howroyd, Jul 31 2018
G.f.: Sum_{k>=0} 3^k * x^(2^k) / (1 + x^(2^k)). - Ilya Gutkovskiy, Dec 14 2020
Dirichlet g.f.: zeta(s)*(2^s-2)/(2^s-3). - Amiram Eldar, Dec 29 2022

A091311 Partial sums of 3^A007814(n).

Original entry on oeis.org

0, 1, 4, 5, 14, 15, 18, 19, 46, 47, 50, 51, 60, 61, 64, 65, 146, 147, 150, 151, 160, 161, 164, 165, 192, 193, 196, 197, 206, 207, 210, 211, 454, 455, 458, 459, 468, 469, 472, 473, 500, 501, 504, 505, 514, 515, 518, 519, 600, 601, 604, 605, 614, 615, 618, 619

Offset: 0

Ralf Stephan, Feb 24 2004

First differences are A061393(n)-1.

Cf. A005836.

Join[{0},Accumulate[3^IntegerExponent[Range[64],2]]] (* Harvey P. Dale, May 20 2025 *)

PARI
```
a(n)=sum(k=1,n,3^valuation(k,2))
```

a(n)=if(n<1,0,if(n%2==0,3*a(n/2)+n/2,3*a((n-1)/2)+(n+1)/2))

def A091311(n): return (int(bin(n)[2:],3)<<1)-n # Chai Wah Wu, Jul 07 2022

Recurrence: a(2n) = 3a(n) + n, a(2n+1) = 3a(n) + n + 1.
G.f.: 1/(1-x) * sum(k>=0, 3^k*t/(1-t^2), t=x^2^k).
a(n) = 2*A005836(n+1) - n.

cp's OEIS Frontend

A191106 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x-2 and 3x are in a.

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A061392 a(n) = a(floor(n/3)) + a(ceiling(n/3)) with a(0) = 0 and a(1) = 1.

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A236313 Recurrence: a(2n) = 3a(n)-1, a(2n+1) = 1.

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A091311 Partial sums of 3^A007814(n).

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