A289814 - OEIS

%I #33 May 09 2024 03:12:18

%S 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,

%T 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,

%U 8,9,10,10,11,12,12,13,12,12,13,14,14,15,0

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

%C 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).

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

%H Rémy Sigrist, <a href="/A289814/b289814.txt">Table of n, a(n) for n = 0..6560</a>

%F a(0) = 0.

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

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

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

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

%F A053735(n) = A000120(A289813(n)) + 2*A000120(a(n)). - _Antti Karttunen_, Jul 20 2017

%e The first values, alongside the ternary representation of n, and the binary representation of a(n), are:

%e n       a(n)    ternary(n)  binary(a(n))

%e --      ----    ----------  ------------

%e 0       0       0           0

%e 1       0       1           0

%e 2       1       2           1

%e 3       0       10          0

%e 4       0       11          0

%e 5       1       12          1

%e 6       2       20          10

%e 7       2       21          10

%e 8       3       22          11

%e 9       0       100         0

%e 10      0       101         0

%e 11      1       102         1

%e 12      0       110         0

%e 13      0       111         0

%e 14      1       112         1

%e 15      2       120         10

%e 16      2       121         10

%e 17      3       122         11

%e 18      4       200         100

%e 19      4       201         100

%e 20      5       202         101

%e 21      4       210         100

%e 22      4       211         100

%e 23      5       212         101

%e 24      6       220         110

%e 25      6       221         110

%e 26      7       222         111

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

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

%o (PARI) a(n) = fromdigits(digits(n, 3)\2, 2); \\ _Ruud H.G. van Tol_, May 08 2024

%o (Python)

%o from sympy.ntheory.factor_ import digits

%o def a(n):

%o     d = digits(n, 3)[1:]

%o     return int("".join('1' if i == 2 else '0' for i in d), 2)

%o print([a(n) for n in range(101)]) # _Indranil Ghosh_, Jul 20 2017

%Y Cf. A000120, A053735, A289813.

%K nonn,base,look

%O 0,7

%A _Rémy Sigrist_, Jul 12 2017