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A289813
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A binary encoding of the ones in ternary representation of n (see Comments for precise definition).
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46
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0, 1, 0, 2, 3, 2, 0, 1, 0, 4, 5, 4, 6, 7, 6, 4, 5, 4, 0, 1, 0, 2, 3, 2, 0, 1, 0, 8, 9, 8, 10, 11, 10, 8, 9, 8, 12, 13, 12, 14, 15, 14, 12, 13, 12, 8, 9, 8, 10, 11, 10, 8, 9, 8, 0, 1, 0, 2, 3, 2, 0, 1, 0, 4, 5, 4, 6, 7, 6, 4, 5, 4, 0, 1, 0, 2, 3, 2, 0, 1, 0, 16
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OFFSET
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0,4
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COMMENTS
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The ones in the binary representation of a(n) correspond to the ones in the ternary representation of n; for example: ternary(42) = 1120 and binary(a(42)) = 1100 (a(42) = 12).
See A289814 for the sequence encoding the twos in ternary representation of n.
By design, a(n) AND A289814(n) = 0 (where AND stands for the bitwise AND operator).
For each pair of numbers without common bits in base 2 representation, say x and y, there is a unique index, say n, such that a(n) = x and A289814(n) = y; in fact, n = A289869(x,y).
The scatterplot of this sequence vs A289814 looks like a Sierpinski triangle pivoted to the side.
For any t > 0: we can adapt the algorithm used here and in A289814 in order to uniquely enumerate every tuple of t numbers mutually without common bits in base 2 representation.
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LINKS
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FORMULA
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a(0) = 0.
a(3*n) = 2 * a(n).
a(3*n+1) = 2 * a(n) + 1.
a(3*n+2) = 2 * a(n).
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EXAMPLE
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The first values, alongside the ternary representation of n, and the binary representation of a(n), are:
n a(n) ternary(n) binary(a(n))
-- ---- ---------- ------------
0 0 0 0
1 1 1 1
2 0 2 0
3 2 10 10
4 3 11 11
5 2 12 10
6 0 20 0
7 1 21 1
8 0 22 0
9 4 100 100
10 5 101 101
11 4 102 100
12 6 110 110
13 7 111 111
14 6 112 110
15 4 120 100
16 5 121 101
17 4 122 100
18 0 200 0
19 1 201 1
20 0 202 0
21 2 210 10
22 3 211 11
23 2 212 10
24 0 220 0
25 1 221 1
26 0 222 0
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MATHEMATICA
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Table[FromDigits[#, 2] &[IntegerDigits[n, 3] /. 2 -> 0], {n, 0, 81}] (* Michael De Vlieger, Jul 20 2017 *)
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PROG
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(PARI) a(n) = my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2)
(Python)
from sympy.ntheory.factor_ import digits
def a(n):
d = digits(n, 3)[1:]
return int("".join('1' if i==1 else '0' for i in d), 2)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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