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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).

%I A084188 #48 Aug 25 2024 16:53:34
%S A084188 1,2,5,11,22,45,90,181,362,724,1448,2896,5792,11585,23170,46340,92681,
%T A084188 185363,370727,741455,1482910,2965820,5931641,11863283,23726566,
%U A084188 47453132,94906265,189812531,379625062,759250124
%N 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).
%C A084188 Numerators in approximation sqrt(2) ~ a(n)/2^n.
%C A084188 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
%C A084188 Largest k such that k^2 <= 2^(2n + 1). - _Irina Gerasimova_, Jul 07 2013
%H A084188 Reinhard Zumkeller, <a href="/A084188/b084188.txt">Table of n, a(n) for n = 0..1000</a>
%F A084188 a(n) = floor(sqrt(2)*2^n).
%F A084188 a(n) = A017910(2*n+1). - _Peter Luschny_, Sep 20 2011
%p A084188 A084188 := n->floor(sqrt(2)*2^n); # _Peter Luschny_, Sep 20 2011
%t A084188 Table[Floor[Sqrt[2] 2^n],{n,0,30}] (* _Harvey P. Dale_, Aug 15 2013 *)
%o A084188 (PARI) a(n)=floor(sqrt(2)<<n) \\ _Charles R Greathouse IV_, Sep 22 2011
%o A084188 (Haskell)
%o A084188 a084188 n = a084188_list !! n
%o A084188 a084188_list = scanl1 (\u v -> 2 * u + v) a004539_list
%o A084188 -- _Reinhard Zumkeller_, Dec 16 2013
%o A084188 (Magma) [Isqrt(2^(2*n+1)):n in[0..40]]; // _Jason Kimberley_, Oct 25 2016
%o A084188 (PARI) {a(n) = sqrtint(2*4^n)}; /* _Michael Somos_, Oct 29 2016 */
%o A084188 (Python)
%o A084188 from math import isqrt
%o A084188 def A084188(n): return isqrt(1<<(n<<1)+1) # _Chai Wah Wu_, Jan 24 2024
%Y A084188 Cf. A084185, A084186, A017910.
%K A084188 nonn,easy
%O A084188 0,2
%A A084188 _Ralf Stephan_, May 18 2003