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
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.
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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
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
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.
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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 *)
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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
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
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a084188 n = a084188_list !! n
a084188_list = scanl1 (\u v -> 2 * u + v) a004539_list
-- Reinhard Zumkeller, Dec 16 2013
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[Isqrt(2^(2*n+1)):n in[0..40]]; // Jason Kimberley, Oct 25 2016
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A084188 := n->floor(sqrt(2)*2^n); # Peter Luschny, Sep 20 2011
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Table[Floor[Sqrt[2] 2^n],{n,0,30}] (* Harvey P. Dale, Aug 15 2013 *)
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a(n)=floor(sqrt(2)<Charles R Greathouse IV, Sep 22 2011
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{a(n) = sqrtint(2*4^n)}; /* Michael Somos, Oct 29 2016 */
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from math import isqrt
def A084188(n): return isqrt(1<<(n<<1)+1) # Chai Wah Wu, Jan 24 2024
Showing 1-3 of 3 results.
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