A017911 Powers of sqrt(2) rounded to nearest integer.

1, 1, 2, 3, 4, 6, 8, 11, 16, 23, 32, 45, 64, 91, 128, 181, 256, 362, 512, 724, 1024, 1448, 2048, 2896, 4096, 5793, 8192, 11585, 16384, 23170, 32768, 46341, 65536, 92682, 131072, 185364, 262144, 370728, 524288

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Apart from offset the same as A057048. - T. D. Noe, Apr 27 2003

Indeed, write the natural numbers as triangle, [1; 2, 3; 4, 5, 6; ...], then the last number in each row is T(n) = n(n+1)/2 = A000217(n), and 2^k is located in the row n with n(n-1)/2 < 2^k <= n(n+1)/2 <=> n^2 - n < 2^(k+1) <= n^2 + n, which means that n = round(sqrt(2^(k+1))). - M. F. Hasler, Feb 20 2012

The rounded curvature of circle in square inscribing or the rounded radius of circle in square circumscribing with initial circle radius = 1 for both cases, see illustration in link. - Kival Ngaokrajang, Aug 07 2013

Even-indexed terms are powers of 2.

Examples

			sqrt(2)^3 = 2.82842712474619..., so a(3) = 3.
sqrt(2)^4 = 4, so a(4) = 4.
sqrt(2)^5 = 5.6568542494923801952..., so a(5) = 6.
sqrt(2)^6 = 8, so a(6) = 8.
sqrt(2)^7 = 11.31370849898476..., so a(7) = 11.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..5000
• Kival Ngaokrajang, Illustration of initial terms of square inscribing
• Eric Weisstein's World of Mathematics, Polygon Inscribing

Crossrefs

Cf. A000079, A057048.

Apart from offset, first differences of A001521.

Programs

Magma

[Round(Sqrt(2)^n): n in [0..40]]; // Vincenzo Librandi, Nov 19 2011

Mathematica

Floor[(Sqrt[2]^Range[0, 40] + 1/2)] (* Vincenzo Librandi, Nov 19 2011 *)

PARI

a(n)=round(sqrt(2)^n) \\ Charles R Greathouse IV, Nov 18 2011

Python

from math import isqrt
def A017911(n): return -isqrt(m:=1<Chai Wah Wu, Jun 18 2024