A084187 First occurrence of exactly n 0's in the binary expansion of sqrt(2).

2, 15, 63, 58, 9, 1003, 524, 454, 1303, 5335, 22472, 8882, 37469, 32279, 220311, 92988, 698343, 24002, 574131, 3333660, 5940559, 4079882, 8356569, 115885798, 76570753, 202460870, 1034477781, 457034356, 1005210009, 3753736439, 2204906858, 50747186116, 32242071604, 159423417084, 114244391078, 74632918239

Offset: 1

Author

Ralf Stephan, May 18 2003

Keywords

• nonn
• hard

Examples

			The binary expansion of sqrt(2) is 1.0110101000001..(A004539) and at position 9, there are five 0's, framed by 1's, so a(5)=9.
		

Crossrefs

Cf. A084185, A084186.

Cf. A233836.

Programs

C

See Links section of A084186.

Mathematica

With[{d=RealDigits[Sqrt[2],2,116*10^6][[1]]},Flatten[Table[SequencePosition[d,Join[ {1},PadRight[{},n,0],{1}],1][[All,1]],{n,25}]]]+1 (* Harvey P. Dale, Dec 12 2022 *)

Python

from math import isqrt
from itertools import count
def A084187(n):
    a, b = 2, (1<>1)|1
    for k in count(1-n):
        if isqrt(a)&b==c:
            return k
        a<<=2 # Chai Wah Wu, Jan 25 2024

Extensions

More terms from Ryan Propper, May 09 2006

a(26)-a(29) from Chai Wah Wu, Jan 25 2024

a(30)-a(36) from Nick Hobson, Feb 15 2024