A338086 -id:A338086 - cp's OEIS frontend

A001196 Double-bitters: only even length runs in binary expansion.

0, 3, 12, 15, 48, 51, 60, 63, 192, 195, 204, 207, 240, 243, 252, 255, 768, 771, 780, 783, 816, 819, 828, 831, 960, 963, 972, 975, 1008, 1011, 1020, 1023, 3072, 3075, 3084, 3087, 3120, 3123, 3132, 3135, 3264, 3267, 3276, 3279, 3312, 3315, 3324, 3327, 3840, 3843

Offset: 0

N. J. A. Sloane, based on an email from Bart la Bastide (bart(AT)xs4all.nl)

Numbers whose set of base 4 digits is {0,3}. - Ray Chandler, Aug 03 2004
n such that there exists a permutation p_1, ..., p_n of 1, ..., n such that i + p_i is a power of 4 for every i. - Ray Chandler, Aug 03 2004
The first 2^n terms of the sequence could be obtained using the Cantor-like process for the segment [0, 4^n-1]. E.g., for n=1 we have [0, {1, 2}, 3] such that numbers outside of braces are the first 2 terms of the sequence; for n=2 we have [0, {1, 2}, 3, {4, 5, 6, 7, 8, 9, 10, 11}, 12, {13, 14}, 15] such that the numbers outside of braces are the first 4 terms of the sequence, etc. - Vladimir Shevelev, Dec 17 2012
From Emeric Deutsch, Jan 26 2018: (Start)
Also, the indices of the compositions having only even parts. For the definition of the index of a composition, see A298644. For example, 195 is in the sequence since its binary form is 11000011 and the composition [2,4,2] has only even parts. 132 is not in the sequence since its binary form is 10000100 and the composition [1,4,1,2] also has odd parts.
The command c(n) from the Maple program yields the composition having index n. (End)
After the k-th step of generating the Koch snowflake curve, label the edges of the curve consecutively 0..3*4^k-1 starting from a vertex of the originating triangle. a(0), a(1), ... a(2^k-1) are the labels of the edges contained in one edge of the originating triangle. Add 4^k to each label to get the labels for the next edge of the triangle. Compare with A191108 in respect of the Sierpinski arrowhead curve. - Peter Munn, Aug 18 2019

Sean A. Irvine, Table of n, a(n) for n = 0..10000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
Ralf Stephan, Table of generating functions. [ps file]
Ralf Stephan, Table of generating functions. [pdf file]
Eric Weisstein's World of Mathematics, Koch Snowflake.
Wikipedia, Koch snowflake.
Index entries for sequences related to binary expansion of n

3 times the Moser-de Bruijn sequence A000695.
Two digits in other bases: A005823, A097252-A097262.
Digit duplication in other bases: A338086, A338754.
Main diagonal of A054238.
Cf. A191108.

int a_next(int a_n) { int t = a_n << 1; return a_n ^ t ^ (t + 3); } /* Falk Hüffner, Jan 24 2022 */

a001196 n = if n == 0 then 0 else 4 * a001196 n' + 3 * b
            where (n',b) = divMod n 2
-- Reinhard Zumkeller, Feb 21 2014

Runs := proc (L) local j, r, i, k: j := 1: r[j] := L[1]: for i from 2 to nops(L) do if L[i] = L[i-1] then r[j] := r[j], L[i] else j := j+1: r[j] := L[i] end if end do: [seq([r[k]], k = 1 .. j)] end proc: RunLengths := proc (L) map(nops, Runs(L)) end proc: c := proc (n) ListTools:-Reverse(convert(n, base, 2)): RunLengths(%) end proc: A := {}: for n to 3350 do if type(product(1+c(n)[j], j = 1 .. nops(c(n))), odd) = true then A := `union`(A, {n}) else  end if end do: A; # most of the Maple  program is due to W. Edwin Clark. - Emeric Deutsch, Jan 26 2018
# second Maple program:
a:= proc(n) option remember;
     `if`(n<2, 3*n, 4*a(iquo(n, 2, 'r'))+3*r)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 24 2022

fQ[n_] := Union@ Mod[Length@# & /@ Split@ IntegerDigits[n, 2], 2] == {0}; Select[ Range@ 10000, fQ] (* Or *)
fQ[n_] := Union@ Join[IntegerDigits[n, 4], {0, 3}] == {0, 3}; Select[ Range@ 10000, fQ] (* Robert G. Wilson v, Dec 24 2012 *)

a(n) = 3*fromdigits(binary(n),4); \\ Kevin Ryde, Nov 07 2020

def inA001196(n):
    while n != 0:
        if n%4 == 1 or n%4 == 2:
            return 0
        n = n//4
    return 1
n, a = 0, 0
while n < 20:
    if inA001196(a):
        print(n,a)
        n = n+1
    a = a+1 # A.H.M. Smeets, Aug 19 2019

from itertools import groupby
def ok2lb(n):
  if n == 0: return True  # by convention
  return all(len(list(g))%2 == 0 for k, g in groupby(bin(n)[2:]))
print([i for i in range(3313) if ok2lb(i)]) # Michael S. Branicky, Jan 04 2021

def A001196(n): return 3*int(bin(n)[2:],4) # Chai Wah Wu, Aug 21 2023

a(2n) = 4*a(n), a(2n+1) = 4*a(n) + 3.
a(n) = 3 * A000695(n).
Sum_{n>=1} 1/a(n) = 0.628725478158702414849086504025451177643560169366348272891020450593453403709... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 12 2022

A037314 Numbers whose base-3 and base-9 expansions have the same digit sum.

Original entry on oeis.org

0, 1, 2, 9, 10, 11, 18, 19, 20, 81, 82, 83, 90, 91, 92, 99, 100, 101, 162, 163, 164, 171, 172, 173, 180, 181, 182, 729, 730, 731, 738, 739, 740, 747, 748, 749, 810, 811, 812, 819, 820, 821, 828, 829, 830, 891, 892, 893, 900, 901, 902, 909, 910, 911

Offset: 0

Clark Kimberling

a(n) = Sum_{i=0..m} d(i)*9^i, where Sum_{i=0..m} d(i)*3^i is the base-3 representation of n.
Numbers that can be written using only digits 0, 1 and 2 in base 9. Also, write n in base 3, read as base 9: (3) [n] (9) in base change notation. a(3n+k) = 9a(n)+k for k in {0,1,2}. - Franklin T. Adams-Watters, Jul 24 2006
Also, every term k corresponds to a unique pair i,j with k = a(i) + 3*a(j) (similarly to the Moser-de Bruijn sequence). - Luis Rato, May 02 2024

Cf. A007089, A208665, A338086 (ternary digit duplication).

Cf. A053735, A053830.

function a(n)
    m, r, b = n, 0, 1
    while m > 0
        m, q = divrem(m, 3)
        r += b * q
        b *= 9
    end
r end
[a(n) for n in 0:53] |> println # Peter Luschny, Jan 03 2021

Table[FromDigits[RealDigits[n, 3], 9], {n, 1, 100}] (* Clark Kimberling, Aug 14 2012 *)
Select[Range[0,1000],Total[IntegerDigits[#,3]]==Total[IntegerDigits[#,9]]&] (* Harvey P. Dale, Feb 17 2020 *)

a(n) = {my(d = digits(n, 3)); subst(Pol(d), x, 9);} \\ Michel Marcus, Apr 09 2015

G.f. f(x) = Sum_{j>=0} 9^j*x^(3^j)*(1+x^(3^j)-2*x^(2*3^j))/((1-x)*(1-x^(3^(j+1)))) satisfies f(x) = 9*(x^2+x+1)*f(x^3) + x*(1+2*x)/(1-x^3). - Robert Israel, Apr 13 2015

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 08 2007

Offset changed to 0 by Clark Kimberling, Aug 14 2012

A338754 Duplicate each decimal digit of n, so 0 -> 00, ..., 9 -> 99.

Original entry on oeis.org

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 1100, 1111, 1122, 1133, 1144, 1155, 1166, 1177, 1188, 1199, 2200, 2211, 2222, 2233, 2244, 2255, 2266, 2277, 2288, 2299, 3300, 3311, 3322, 3333, 3344, 3355, 3366, 3377, 3388, 3399, 4400, 4411, 4422, 4433, 4444, 4455, 4466

Offset: 0

Kevin Ryde, Nov 06 2020

This is equivalent to changing decimal digits 0,1,..,9 to base 100 digits 0,11,..,99, so the sequence is numbers which can be written in base 100 using only digits 0,11,..,99. Also, numbers whose decimal digit runs are all even lengths (including 0 as no digits at all).

This sequence first differs from A044836 (apart from term 0) at a(100) = 110000 whereas A044836(100) = 10011, because A044836 allows odd length digit runs provided there are more even than odd.

			For n=5517, digits duplicate to a(n) = 55551177.

Kevin Ryde, Table of n, a(n) for n = 0..10000

Cf. A051022 (0 above each digit), A044836.

Other bases: A001196, A338086.

PARI
```
a(n) = fromdigits(digits(n),100)*11;
```

def A338754(n): return int(''.join(d*2 for d in str(n))) # Chai Wah Wu, May 07 2022

a(n) = Sum_{i=0..k} 11*d[i]*100^i where the decimal expansion of n is n = Sum_{i=0..k} d[i]*10^i with digits 0 <= d[i] <= 9.

a(n) = A051022(n)*11 for n > 0. - Kritsada Moomuang, Oct 20 2019

A208665 Numbers that match odd ternary polynomials; see Comments.

Original entry on oeis.org

3, 6, 27, 30, 33, 54, 57, 60, 243, 246, 249, 270, 273, 276, 297, 300, 303, 486, 489, 492, 513, 516, 519, 540, 543, 546, 2187, 2190, 2193, 2214, 2217, 2220, 2241, 2244, 2247, 2430, 2433, 2436, 2457, 2460, 2463, 2484, 2487, 2490, 2673, 2676, 2679

Offset: 1

Clark Kimberling, Feb 29 2012

The ternary polynomials (having all coefficients in {0,1,2}) are enumerated at A207966. This sequence shows the numbers n for which p(n,-x)=-p(n,x).

Cf. A037314, A207966, A338086 (ternary digit duplication).

t = Table[IntegerDigits[n, 3], {n, 1, 4000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]
Table[p[n, x], {n, 1, 30}]
even = {}; Do[n++; If[(p[n, x] /. x -> -x) == p[n, x], AppendTo[even, n]], {n, 1600}];
even     (* A037314 for n >= 2 *)
odd = {}; Do[n++; If[(p[n, x] /. x -> -x) == -p[n, x], AppendTo[odd, n]], {n, 3900}];
odd      (* A208665 *)

a(n) = 3*fromdigits(digits(n,3),9); \\ Kevin Ryde, Oct 17 2020

A163343 Central diagonal of A163334 and A163336.

Original entry on oeis.org

0, 4, 8, 44, 40, 36, 72, 76, 80, 404, 400, 396, 360, 364, 368, 332, 328, 324, 648, 652, 656, 692, 688, 684, 720, 724, 728, 3644, 3640, 3636, 3600, 3604, 3608, 3572, 3568, 3564, 3240, 3244, 3248, 3284, 3280, 3276, 3312, 3316, 3320, 2996, 2992, 2988

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

It is easy to see by induction that these terms are always divisible by 4.

Cf. A163344 (quarter), A128173, A163332, A163334, A338086.

Peano curve axes: A163480, A163481.

a(n) = my(v=digits(n,3),s=Mod(0,2)); for(i=1,#v, if(s,v[i]=2-v[i]); s+=v[i]); fromdigits(v,9)<<2; \\ Kevin Ryde, Nov 06 2020

a(n) = 4*A163344(n).

a(n) = A163332(A338086(n)) = A338086(A128173(n)). - Kevin Ryde, Nov 06 2020

A342218 The n-th and a(n)-th points of the Peano curve (A163528, A163529) are symmetrical with respect to the line X=Y.

Original entry on oeis.org

0, 5, 6, 7, 4, 1, 2, 3, 8, 45, 50, 51, 52, 49, 46, 47, 48, 53, 54, 59, 60, 61, 58, 55, 56, 57, 62, 63, 68, 69, 70, 67, 64, 65, 66, 71, 36, 41, 42, 43, 40, 37, 38, 39, 44, 9, 14, 15, 16, 13, 10, 11, 12, 17, 18, 23, 24, 25, 22, 19, 20, 21, 26, 27, 32, 33, 34, 31

Offset: 0

Rémy Sigrist, Mar 05 2021

In other words, a(n) is the unique k such that A163528(n) = A163529(k) and A163528(k) = A163529(n).

This sequence is a self-inverse permutation of the nonnegative integers.

			The Peano curve (A163528, A163529) begins as follows:
       +-----+-----+
       |6     7     8
       |
       +-----+-----+
        5     4    |3
                   |
       +-----+-----+
        0     1     2
- so a(0) = 0,
     a(1) = 5,
     a(2) = 6,
     a(3) = 7,
     a(4) = 4,
     a(8) = 8.

Rémy Sigrist, Table of n, a(n) for n = 0..6560
Rémy Sigrist, PARI program for A342218
Index entries for sequences that are permutations of the nonnegative integers

See A342217 and A342224 for similar sequences.

Cf. A163528, A163529, A338086.

PARI
```
See Links section.
```

my(table=[0,5,6,7,4,1,2,3,8]); a(n) = fromdigits(apply(d->table[d+1], digits(n,9)), 9); \\ Kevin Ryde, Mar 07 2021

a(n) = n iff n belongs to A338086.

a(n) < 9^k for any n < 9^k.

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A001196 Double-bitters: only even length runs in binary expansion.

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A037314 Numbers whose base-3 and base-9 expansions have the same digit sum.

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A338754 Duplicate each decimal digit of n, so 0 -> 00, ..., 9 -> 99.

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A208665 Numbers that match odd ternary polynomials; see Comments.

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A163343 Central diagonal of A163334 and A163336.

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A342218 The n-th and a(n)-th points of the Peano curve (A163528, A163529) are symmetrical with respect to the line X=Y.

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