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%I #51 May 19 2024 02:09:23
%S 1,1,2,2,4,5,8,11,16,22,32,45,64,90,128,181,256,362,512,724,1024,1448,
%T 2048,2896,4096,5792,8192,11585,16384,23170,32768,46340,65536,92681,
%U 131072,185363,262144,370727,524288
%N Powers of sqrt(2) rounded down.
%C a(n) is the number of positive squares <= 2^n (cf. A136417). - _Hans Havermann_, Apr 05 2008
%C If expressed to two significant digits, these are the f-stop numbers in photography: 1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22, ...
%C There are also "half stops" (sqrt(2)^(n/2)) and "third stops" (sqrt(2)^(n/3)): 1, 1.4, 1.6, 1.8, 2.0, 2.2, 2.5, 2.8, 3.2, 3.6, 4, 4.5, 5, 5.7, 6.3, 7.1, 8, 9, 10.
%C a(n) is also the ratio (rounded down) of the curvature of the circle inscribed in the n-th 45-45-90 triangle to that of the circle inscribed in the 1st triangle, with the triangles arranged in a spiral as shown in the illustration in the links section. - _Kival Ngaokrajang_, Aug 28 2013
%C a(n) is also the total length of Heighway dragon (rounded down) after n-iterations when L(0) = 1. See illustration in links. - _Kival Ngaokrajang_, Dec 15 2013
%H Vincenzo Librandi, <a href="/A017910/b017910.txt">Table of n, a(n) for n = 0..1000</a>
%H Kival Ngaokrajang, <a href="/A017910/a017910.pdf">Illustration of initial terms</a>.
%H Kival Ngaokrajang, <a href="/A017910/a017910_1.pdf">Illustration of Heighway dragon for n = 0..5</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Dragon_curve">Dragon curve</a>.
%F a(n) = A000196(A000079(n)). - _Jason Kimberley_, Oct 28 2016
%F a(n) = A017912(n)-1 if n is odd. a(n) = A017912(n) = 2^(n/2) if n is even. - _Chai Wah Wu_, Jul 26 2022
%p A017910 := n->floor(sqrt(2^n)); # _Peter Luschny_, Sep 20 2011
%t Floor[(Sqrt[2])^Range[0,40]] (* _Vincenzo Librandi_, Nov 20 2011 *)
%o (PARI) a(n)=sqrtint(2^n) \\ _Charles R Greathouse IV_, Sep 22 2011
%o (Magma) [Floor(Sqrt(2^n)): n in [0..40]]; // _Vincenzo Librandi_, Nov 20 2011
%o (Magma) [Isqrt(2^n):n in[0..40]]; // _Jason Kimberley_, Oct 25 2016
%o (Python)
%o from math import isqrt
%o def A017910(n): return isqrt(1<<n) if n&1 else 1<<(n>>1) # _Chai Wah Wu_, Jul 26 2022
%Y Cf. A136417, A017912. Bisections: A000079, A084188.
%Y Partial sums of A190568.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
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