A272170 Second most significant bit of Fibonacci numbers > 1 written in base 2.

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

Offset

3,1

Comments

It is conjectured that there are no more than two consecutive "0's" or “1’s” (tested up to n=10^5). The sequence looks quasiperiodic and its Fourier spectrum seems to have a fractal structure.

Links

Chai Wah Wu, Table of n, a(n) for n = 3..10000

Formula

a(n) = floor(A000045(n)/(2^(ceiling(log_2(A000045(n) + 1)) - 2))) - 2.

a(n) = A079944(A000045(n)-2). - Michel Marcus, Apr 22 2016

Example

(second MSB in parenthesis)
  n   A000045(n)      A004685(n)
  3      2       ->   1(0)
  4      3       ->   1(1)
  5      5       ->   1(0)1
  6      8       ->   1(0)00
  7      13      ->   1(1)01
  8      21      ->   1(0)101
  9      34      ->   1(0)0010
  10     55      ->   1(1)0111
...

Mathematica

nmax = 120; Table[IntegerDigits[Fibonacci[j], 2][[2]], {j, 3, nmax}]

Prog

(PARI) a(n) = binary(fibonacci(n))[2]; \\ Michel Marcus, Apr 25 2016
(Python)
A272170_list, a, b = [], 1 , 1
for n in range(3, 10001):
    a, b = b, a+b
    A272170_list.append(int(bin(b)[3])) # Chai Wah Wu, Feb 07 2018

Crossrefs

Cf. A000045, A004685, A079944, A271591.

Sequence in context: A163584 A353307 A286655 * A126565 A215531 A305386

Adjacent sequences: A272167 A272168 A272169 * A272171 A272172 A272173

Keyword

nonn,base

Author

Andres Cicuttin, Apr 21 2016

Status

approved