A196168 In binary representation of n: replace each 0 with 1, and each 1 with 10.

1, 2, 5, 10, 11, 22, 21, 42, 23, 46, 45, 90, 43, 86, 85, 170, 47, 94, 93, 186, 91, 182, 181, 362, 87, 174, 173, 346, 171, 342, 341, 682, 95, 190, 189, 378, 187, 374, 373, 746, 183, 366, 365, 730, 363, 726, 725, 1450, 175, 350, 349, 698, 347, 694, 693, 1386

Offset

0,2

Comments

All terms are numbers with no two adjacent zeros in binary representation, cf. A003754;

a(odd) = even and a(even) = odd;

A023416(a(n)) <= A000120(a(n)), equality iff n = 2^k - 1 for k > 0;

A055010(n+1) = A196168(A000079(n));

A000120(a(n)) = A070939(n);

A023416(a(n)) = A000120(n);

A070939(a(n)) = A070939(n) + A000120(n).

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Index entries for sequences related to binary expansion of n

Formula

n = Sum_{i=0..1} b(i)*2^i with 0 <= b(i) <= 1, L >= 0, then a(n) = h(0,L) with h(v,i) = if i > L then v, otherwise h((2*v+1)*(b(i)+1),i-1).

From Jeffrey Shallit, Oct 28 2021: (Start)

a(n) satisfies the recurrences:

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

a(4n) = -2*a(n) + 3*a(2n)

a(8n+2) = -8*a(n) + 8*a(2n) + a(4n+2)

a(8n+6) = -4*a(2n) + 5*a(4n+2)

which shows that a(n) is a 2-regular sequence. (End)

Example

n =  7 ->  111 ->  101010 ->  a(7) = 42;
n =  8 -> 1000 ->   10111 ->  a(8) = 23;
n =  9 -> 1001 ->  101110 ->  a(9) = 46;
n = 10 -> 1010 ->  101101 -> a(10) = 45;
n = 11 -> 1011 -> 1011010 -> a(11) = 90;
n = 12 -> 1100 ->  101011 -> a(12) = 43.

Mathematica

Table[FromDigits[Flatten[IntegerDigits[n, 2]/.{{0->1, 1->{1, 0}}}], 2], {n, 0, 120}] (* Harvey P. Dale, Dec 12 2017 *)

Prog

(Haskell)
import Data.List (unfoldr)
a196168 0 = 1
a196168 n = foldl (\v b -> (2 * v + 1)*(b + 1)) 0 $ reverse $ unfoldr
   (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2) n
   where r v b = (2 * v + 1)*(b+1)
(Python)
def a(n):
    b = bin(n)[2:]
    return int(b.replace('1', 't').replace('0', '1').replace('t', '10'), 2)
print([a(n) for n in range(56)]) # Michael S. Branicky, Oct 28 2021

Crossrefs

Cf. A179888, A005614.

Sequence in context: A240032 A187792 A176356 * A018514 A018288 A245480

Adjacent sequences: A196165 A196166 A196167 * A196169 A196170 A196171

Keyword

nonn

Author

Reinhard Zumkeller, Oct 28 2011

Status

approved