A084186 -id:A084186 - cp's OEIS frontend

A233836 Run lengths of ones and zeros in binary expansion of sqrt(2), cf. A004539.

1, 1, 2, 1, 1, 1, 1, 5, 1, 2, 4, 2, 2, 2, 2, 2, 7, 2, 3, 1, 4, 2, 2, 2, 1, 2, 1, 4, 1, 3, 1, 1, 2, 2, 1, 1, 5, 1, 2, 3, 1, 2, 2, 1, 2, 2, 2, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 5, 1, 1, 2, 5, 3, 3, 1, 1, 1, 2, 1, 4, 1, 2, 5, 1, 1, 3, 1, 1, 1, 1, 3

Offset: 1

Reinhard Zumkeller, Dec 16 2013

nonn

			. A004539: 1 0 1 1 0 1 0 1 0 0 0 0 0 1 0 0 1 1 1 1 0 0 1 1 0 0 1 1 ...
.    runs: _ _ ___ _ _ _ _ _________ _ ___ _______ ___ ___ ___ ___ ...
. lengths: 1 1  2  1 1 1 1     5     1  2     4     2   2   2   2  ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A084186, A084187, A175911.

import Data.List (group)
a233836 n = a233836_list !! (n-1)
a233836_list = map length $ group a004539_list

Length/@Split[RealDigits[Sqrt[2],2,300][[1]]] (* Harvey P. Dale, Oct 11 2020 *)

A084185 First occurrence of binary n in the binary expansion of sqrt(2).

Original entry on oeis.org

1, 1, 3, 8, 1, 3, 17, 8, 14, 4, 1, 19, 3, 18, 17, 8, 62, 51, 14, 6, 4, 1, 42, 80, 19, 3, 41, 18, 40, 17, 31, 8, 57, 62, 128, 51, 69, 20, 14, 6, 104, 4, 111, 66, 1, 96, 42, 146, 80, 50, 19, 118, 3, 77, 41, 126, 18, 98, 40, 17, 75, 33, 31, 453, 8, 57

Offset: 1

Ralf Stephan, May 18 2003

a(n)=1 iff n in A084188. a(2^(n+1)+1) = A084187(n)-1.

			The binary expansion of sqrt(2) is 1.0110101000001..(A004539) and binary(10)=1010 occurs first at index 4, so a(10)=4.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A084186.

With[{s2=RealDigits[Sqrt[2],2,1000][[1]],n2=IntegerDigits[n,2]}, Flatten[ Table[First[ Position[Partition[s2,Length[n2],1],n2]],{n,70}]]] (* Harvey P. Dale, Oct 11 2012 *)

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

Original entry on oeis.org

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

Ralf Stephan, May 18 2003

			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.

Cf. A084185, A084186.

Cf. A233836.

C
```
See Links section of A084186.
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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 *)

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

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

A084188 a(0)=1, a(n+1) = 2*a(n) + b(n+2), where b(n)=A004539(n) is the n-th bit in the binary expansion of sqrt(2).

Original entry on oeis.org

1, 2, 5, 11, 22, 45, 90, 181, 362, 724, 1448, 2896, 5792, 11585, 23170, 46340, 92681, 185363, 370727, 741455, 1482910, 2965820, 5931641, 11863283, 23726566, 47453132, 94906265, 189812531, 379625062, 759250124

Offset: 0

Ralf Stephan, May 18 2003

Numerators in approximation sqrt(2) ~ a(n)/2^n.
a(n) is the number k such that {log_2(k)} < 1/2 < {log_2(k+1)}, where { } = fractional part.  Equivalently, the jump sequence of f(x) = log_2(x), in the sense that these are the positive integers k for which round(log_2(k)) < round(log_2(k+1)); see A219085. - Clark Kimberling, Jan 01 2013
Largest k such that k^2 <= 2^(2n + 1). - Irina Gerasimova, Jul 07 2013

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A084185, A084186, A017910.

a084188 n = a084188_list !! n
a084188_list = scanl1 (\u v -> 2 * u + v) a004539_list
-- Reinhard Zumkeller, Dec 16 2013

[Isqrt(2^(2*n+1)):n in[0..40]]; // Jason Kimberley, Oct 25 2016

A084188 := n->floor(sqrt(2)*2^n); # Peter Luschny, Sep 20 2011

Table[Floor[Sqrt[2] 2^n],{n,0,30}] (* Harvey P. Dale, Aug 15 2013 *)

a(n)=floor(sqrt(2)<Charles R Greathouse IV, Sep 22 2011

{a(n) = sqrtint(2*4^n)}; /* Michael Somos, Oct 29 2016 */

from math import isqrt
def A084188(n): return isqrt(1<<(n<<1)+1) # Chai Wah Wu, Jan 24 2024

a(n) = floor(sqrt(2)*2^n).

a(n) = A017910(2*n+1). - Peter Luschny, Sep 20 2011

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A233836 Run lengths of ones and zeros in binary expansion of sqrt(2), cf. A004539.

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A084185 First occurrence of binary n in the binary expansion of sqrt(2).

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

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A084188 a(0)=1, a(n+1) = 2*a(n) + b(n+2), where b(n)=A004539(n) is the n-th bit in the binary expansion of sqrt(2).

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