A289814 A binary encoding of the twos in ternary representation of n (see Comments for precise definition).

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

Offset: 0

Author

Rémy Sigrist, Jul 12 2017

Keywords

• nonn
• base
• look

Comments

The ones in the binary representation of a(n) correspond to the twos in the ternary representation of n; for example: ternary(42) = 1120 and binary(a(42)) = 10 (a(42) = 2).

See A289813 for the sequence encoding the ones in ternary representation of n and additional comments.

Examples

			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       0       1           0
2       1       2           1
3       0       10          0
4       0       11          0
5       1       12          1
6       2       20          10
7       2       21          10
8       3       22          11
9       0       100         0
10      0       101         0
11      1       102         1
12      0       110         0
13      0       111         0
14      1       112         1
15      2       120         10
16      2       121         10
17      3       122         11
18      4       200         100
19      4       201         100
20      5       202         101
21      4       210         100
22      4       211         100
23      5       212         101
24      6       220         110
25      6       221         110
26      7       222         111
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..6560

Crossrefs

Cf. A000120, A053735, A289813.

Programs

Mathematica

Table[FromDigits[#, 2] &[IntegerDigits[n, 3] /. d_ /; d > 0 :> d - 1], {n, 0, 81}] (* Michael De Vlieger, Jul 20 2017 *)

PARI

a(n) = my (d=digits(n,3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2)

PARI

a(n) = fromdigits(digits(n, 3)\2, 2); \\ Ruud H.G. van Tol, May 08 2024

Python

from sympy.ntheory.factor_ import digits
def a(n):
    d = digits(n, 3)[1:]
    return int("".join('1' if i == 2 else '0' for i in d), 2)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jul 20 2017

Formula

a(0) = 0.

a(3*n) = 2 * a(n).

a(3*n+1) = 2 * a(n).

a(3*n+2) = 2 * a(n) + 1.

Also, a(n) = A289813(A004488(n)).

A053735(n) = A000120(A289813(n)) + 2*A000120(a(n)). - Antti Karttunen, Jul 20 2017