A084187 -id:A084187 - 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 *)

A084186 First occurrence of exactly n 1's in the binary expansion of sqrt(2).

Original entry on oeis.org

1, 3, 40, 17, 74, 265, 31, 336, 11937, 1403, 8894, 3524, 33223, 126903, 3067, 109312, 390536, 553171, 280266, 962560, 1747112, 1740081, 30793169, 13109551, 118101037, 1077718187, 44908294, 1528865059, 1647265647, 3913429742, 10501492774, 4702573600, 81557258556, 107498528405

Offset: 1

Ralf Stephan, May 18 2003

			The binary expansion of sqrt(2) is 1.0110101000001..(A004539) and at position 17, there are four 1's, framed by 0's, so a(4)=17.

Nick Hobson, C program

Cf. A084185, A084187.

Cf. A233836.

C
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See Links section.
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from itertools import count
from math import isqrt
def A084186(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 24 2024

a(21)-a(29) from Chai Wah Wu, Jan 25 2024

a(30)-a(34) from Nick Hobson, Feb 15 2024

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 *)

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

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A084186 First occurrence of exactly n 1's in the binary expansion of sqrt(2).

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

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Mathematica