A160381 -id:A160381 - cp's OEIS frontend

A062756 Number of 1's in ternary (base-3) expansion of n.

0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 2, 3, 2, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 3, 4, 3, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 2, 3, 2, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 3, 4, 3, 2, 3, 2, 1, 2, 1, 2, 3, 2

Offset: 0

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 16 2001

Fixed point of the morphism: 0 ->010; 1 ->121; 2 ->232; ...; n -> n(n+1)n, starting from a(0)=0. - Philippe Deléham, Oct 25 2011

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
F. T. Adams-Watters and F. Ruskey, Generating Functions for the Digital Sum and Other Digit Counting Sequences, JIS 12 (2009) 09.5.6.
Michael Gilleland, Some Self-Similar Integer Sequences
S. Northshield, An Analogue of Stern's Sequence for Z[sqrt(2)], Journal of Integer Sequences, 18 (2015), #15.11.6.
Kevin Ryde, Iterations of the Terdragon Curve, see index "dir".
Robert Walker, Self Similar Sloth Canon Number Sequences

Cf. A080846, A343785 (first differences).
Cf. A081606 (indices of !=0).
Indices of terms 0..6: A005823, A023692, A023693, A023694, A023695, A023696, A023697.
Numbers of: A077267 (0's), A081603 (2's), A160384 (1's+2's).
Other bases: A000120, A160381, A268643.

a062756 0 = 0
a062756 n = a062756 n' + m `mod` 2 where (n',m) = divMod n 3
-- Reinhard Zumkeller, Feb 21 2013

Table[Count[IntegerDigits[i, 3], 1], {i, 0, 200}]
Nest[Join[#, # + 1, #] &, {0}, 5] (* IWABUCHI Yu(u)ki, Sep 08 2012 *)

a(n)=if(n<1,0,a(n\3)+(n%3)%2) \\ Paul D. Hanna, Feb 24 2006

a(n)=hammingweight(digits(n,3)%2); \\ Ruud H.G. van Tol, Dec 10 2023

from sympy.ntheory import digits
def A062756(n): return digits(n,3)[1:].count(1) # Chai Wah Wu, Dec 23 2022

a(0) = 0, a(3n) = a(n), a(3n+1) = a(n)+1, a(3n+2) = a(n). - Vladeta Jovovic, Jul 18 2001
G.f.: (Sum_{k>=0} x^(3^k)/(1+x^(3^k)+x^(2*3^k)))/(1-x). In general, the generating function for the number of digits equal to d in the base b representation of n (0 < d < b) is (Sum_{k>=0} x^(d*b^k)/(Sum_{i=0..b-1} x^(i*b^k)))/(1-x). - Franklin T. Adams-Watters, Nov 03 2005 [For d=0, use the above formula with d=b: (Sum_{k>=0} x^(b^(k+1))/(Sum_{i=0..b-1} x^(i*b^k)))/(1-x), adding 1 if you consider the representation of 0 to have one zero digit.]
a(n) = a(floor(n/3)) + (n mod 3) mod 2. - Paul D. Hanna, Feb 24 2006

More terms from Vladeta Jovovic, Jul 18 2001

A292371 A binary encoding of 1-digits in the base-4 representation of n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Sep 15 2017

			   n      a(n)     base-4(n)  binary(a(n))
                  A007090(n)  A007088(a(n))
  --      ----    ----------  ------------
   1        1          1           1
   2        0          2           0
   3        0          3           0
   4        2         10          10
   5        3         11          11
   6        2         12          10
   7        2         13          10
   8        0         20           0
   9        1         21           1
  10        0         22           0
  11        0         23           0
  12        0         30           0
  13        1         31           1
  14        0         32           0
  15        0         33           0
  16        4        100         100
  17        5        101         101
  18        4        102         100

Antti Karttunen, Table of n, a(n) for n = 0..65536
Rémy Sigrist, Interactive scatterplot of (a(n), A292372(n), A292373(n)) for n=0..4^8-1 [provided your web browser supports the Plotly library, you should see icons on the top right corner of the page: if you choose "Orbital rotation", then you will be able to rotate the plot alongside three axes, the 3D plot here corresponds to a Sierpiński triangle-based pyramid]
Index entries for sequences related to binary expansion of n

Cf. A003188, A004198, A007088, A007090, A059905, A160381, A292272, A292370, A292372, A292373.

Cf. A289813 (analogous sequence for base 3).

Table[FromDigits[IntegerDigits[n, 4] /. k_ /; IntegerQ@ k :> If[k == 1, 1, 0], 2], {n, 0, 112}] (* Michael De Vlieger, Sep 21 2017 *)

from sympy.ntheory.factor_ import digits
def a(n):
    k=digits(n, 4)[1:]
    return 0 if n==0 else int("".join('1' if i==1 else '0' for i in k), 2)
print([a(n) for n in range(116)]) # Indranil Ghosh, Sep 21 2017

def A292371(n): return int(bin(n&~(n>>1))[:1:-2][::-1],2) # Chai Wah Wu, Jun 30 2022

a(n) = A059905(A292272(n)) = A059905(n AND A003188(n)), where AND is bitwise-AND (A004198).

For all n >= 0, A000120(a(n)) = A160381(n).

A160383 Number of 3's in base-4 representation of n.

Original entry on oeis.org

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

Offset: 0

Frank Ruskey, Jun 05 2009

F. T. Adams-Watters and F. Ruskey, Generating Functions for the Digital Sum and Other Digit Counting Sequences, JIS 12 (2009) 09.5.6.

Cf. A007090 (base 4), A160380 (0's), A160381 (1's), A160382 (2's).

Cf. A283316 (mod 2).

a(n) = #select(x->(x==3), digits(n, 4)); \\ Michel Marcus, Mar 24 2020

Recurrence relation: a(0) = 0, a(4m+3) = 1+a(m), a(4m) = a(4m+1) = a(4m+2) = a(m).
G.f.: (1/(1-z))*Sum_{m>=0} (z^(3*4^m)*(1 - z^(4^m))/(1 - z^(4^(m+1)))).
Morphism: 0, j -> j,j,j,j+1; e.g., 0 -> 0001 -> 0001000100011112 -> ...

A039005 Numbers whose base-4 representation has the same number of 1's and 3's.

Original entry on oeis.org

0, 2, 7, 8, 10, 13, 19, 27, 28, 30, 32, 34, 39, 40, 42, 45, 49, 52, 54, 57, 67, 75, 76, 78, 95, 99, 107, 108, 110, 112, 114, 119, 120, 122, 125, 128, 130, 135, 136, 138, 141, 147, 155, 156, 158, 160, 162, 167, 168, 170, 173, 177, 180, 182, 185, 193, 196, 198

Offset: 1

Olivier Gérard

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A007090, A160381, A160383.

Cf. A004171 (subsequence).

A329101 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the number of 1's in the base 4 expansion of n equals the number of 2's in the base 4 expansion of a(n).

Original entry on oeis.org

0, 2, 1, 3, 6, 10, 8, 9, 4, 11, 5, 7, 12, 14, 13, 15, 18, 26, 22, 24, 34, 42, 38, 40, 25, 41, 27, 30, 32, 43, 33, 35, 16, 36, 17, 19, 37, 46, 39, 44, 20, 45, 21, 23, 28, 47, 29, 31, 48, 50, 49, 51, 54, 58, 56, 57, 52, 59, 53, 55, 60, 62, 61, 63, 66, 74, 70, 72

Offset: 0

Rémy Sigrist, Nov 07 2019

This sequence is a permutation of the nonnegative integers with inverse A329180.
Apparently, fixed points correspond to A001196.
The sequence has fractal features; for any k >= 0, the set of points { (n, a(n)), n = 0..4^k-1 } is symmetrical relative to the line of equation y + x = 4^k - 1 (see scatterplots in Links section).

			The first terms, alongside the base 4 representations of n and of a(n), are:
  n   a(n)  qua(n)  qua(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     2       1          2
   2     1       2          1
   3     3       3          3
   4     6      10         12
   5    10      11         22
   6     8      12         20
   7     9      13         21
   8     4      20         10
   9    11      21         23
  10     5      22         11
  11     7      23         13
  12    12      30         30
  13    14      31         32
  14    13      32         31
  15    15      33         33

Rémy Sigrist, Table of n, a(n) for n = 0..4095
Rémy Sigrist, PARI program for A329101
Rémy Sigrist, Scatterplot of the sequence for n = 0..4^3-1
Rémy Sigrist, Scatterplot of the sequence for n = 0..4^10-1
Rémy Sigrist, Colored scatterplot of the sequence for n = 0..4^10-1 (where the color is function of A160381(n))
Index entries for sequences that are permutations of the natural numbers

Cf. A001196, A160381, A160382, A329180.

PARI
```
See Links section.
```

A160381(n) = A160382(a(n)).

A335380 a(n) is the X-coordinate of the n-th point of the Kochawave curve; sequence A335381 gives Y-coordinates.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 6, 6, 7, 8, 8, 9, 10, 10, 11, 12, 12, 9, 12, 12, 13, 16, 14, 15, 16, 16, 17, 18, 18, 19, 18, 18, 19, 22, 20, 21, 20, 18, 19, 18, 18, 19, 18, 18, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 24, 24, 25, 26, 26, 27, 28, 28, 29, 30, 30

Offset: 0

Rémy Sigrist, Jun 04 2020

nonn

Coordinates are given on a hexagonal lattice with X-axis and Y-axis as follows:
           Y
          /
         /
        0 ---- X
The Kochawave curve is a variant of the Koch curve that can be built by successively applying the following substitution to an initial vector (1, 0):
                           .+ C
                        .../
                     ...  /
                  ...    /
        +------>+.      +------>+
        A       B       D       E
- the points A, B, D and E are aligned and equally spaced,
- the points D, C and E form an equilateral triangle
  (for the Koch curve, the points B, C and D form an equilateral triangle).
The distance between two consecutive points is related to A160381:
- for any n >= 0, let z(n) = a(n) + A335381(n) * exp(i*Pi/3) (where i denotes the imaginary unit),
- the square of the distance from z(n) to z(n+1) is 3^A160381(n).

			The Kochawave curve starts (on a hexagonal lattice) as follows:
    .       .       .       .       .       .       +       .       .       .
                                                   /|6
                                                  / |
                                                 /  |
        .       .       .       .       .       .   |   .      .+       .       .
                                               /    |       .../ 8
                                              /     |    ...  /
                                             /      | ...    /
    .       .       .       .       .       .       +.      +       .       .
                                           /         7      |9
                                          /                 |
                                         /                  |
        .       .      .+       .      .+       .       +11 |   .      .+       .
                    .../ 2          ...  5             / \  |       .../ 14
                 ...  /          ...                  /   \ |    ...  /
              ...    /        ...                    /     \| ...    /
    +-------+.      +-------+.      .       .       +-------+.      +-------+
     0       1       3       4                       12   13 10      15      16
- hence a(8) = a(9) = a(11) = a(12) = 6.

Rémy Sigrist, Table of n, a(n) for n = 0..4096
Rémy Sigrist, The Kochawave curve, a variant of the Koch curve, arXiv:2210.17320 [math.HO], 2022.
Rémy Sigrist, Representation of the Kochawave curve
Rémy Sigrist, Representation of the first iterations of the Kochawave curve
Rémy Sigrist, PARI program for A335380
Index entries for sequences related to coordinates of 2D curves

Cf. A160381, A335381.

PARI
```
See Links section.
```

a(4^k) = 3^k for any k >= 0.

A374646 Paradiddle version of Thue-Morse sequence.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1

Offset: 0

Robert P. P. McKone, Jul 15 2024

A paradiddle is a basic drum pattern, either "left left right left" or "right right left right". We can take left, right to be either 0, 1 or 1, 0.

Limiting word of the morphism with maps 0 |--> 0100, 1 |--> 1011 and axiom 1011. - Joerg Arndt, Jul 15 2024

			k = 0: Sequence starts at its simplest form;
1.
-----------------------------------------------
k = 1: The 1 of the initial sequence expands following the morphism rules, where 1 -> {1, 0, 1, 1} and 0 -> {0, 1, 0, 0}, resulting in;
1, 0, 1, 1.
-----------------------------------------------
k = 2: Each element of the initial sequence expands following the morphism rules, where 1 -> {1, 0, 1, 1} and 0 -> {0, 1, 0, 0};
1, 0, 1, 1,
0, 1, 0, 0,
1, 0, 1, 1,
1, 0, 1, 1.
-----------------------------------------------
k = 3: The expansion is applied recursively, giving:
1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1,
0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0,
1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1,
1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1.

Robert P. P. McKone, Table of n, a(n) for n = 0..16383
Index entries for sequences that are fixed points of mappings

Cf. A160381, A130198 (single paradiddle), A010059, A010060, A374724.

SubstitutionSystem[{1 -> {1, 0, 1, 1}, 0 -> {0, 1, 0, 0}}, {1}, {4}] // Flatten

first(n,v=[1])=if(n>4*#v, v=first((n+3)\4)); my(u=List()); for(i=1,#v-1, listput(u,v[i]); listput(u,1-v[i]); listput(u,v[i]); listput(u,v[i])); my(t=vector(n-#u,i,if(i==2,1-v[#v],v[#v]))); for(j=1,#t, listput(u,t[j])); Vec(u) \\ Charles R Greathouse IV, Jul 31 2024

a(n) = A160381(n)+1 mod 2. - Kevin Ryde, Dec 28 2024

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A062756 Number of 1's in ternary (base-3) expansion of n.

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A292371 A binary encoding of 1-digits in the base-4 representation of n.

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A160383 Number of 3's in base-4 representation of n.

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A039005 Numbers whose base-4 representation has the same number of 1's and 3's.

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A329101 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the number of 1's in the base 4 expansion of n equals the number of 2's in the base 4 expansion of a(n).

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A335380 a(n) is the X-coordinate of the n-th point of the Kochawave curve; sequence A335381 gives Y-coordinates.

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A374646 Paradiddle version of Thue-Morse sequence.

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