A057048 a(n) = A017911(n+1) = round(sqrt(2)^(n+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, 741455, 1048576

Offset: 0

Author

M. F. Hasler, Feb 20 2012

Keywords

• nonn
• easy

Comments

If the natural numbers A000027 are written as a triangle, then a(n) gives the row of the triangle in which the number 2^n can be found. See A017911 for a more elaborate explanation and relation with A000217. [Original definition by Clark Kimberling, Jul 30 2000, clarified by M. F. Hasler, Feb 20 2012, following an observation from T. D. Noe, Apr 27 2003]

Examples

			Write the natural numbers A000027 as a triangle:
row 1: 1 . . . <- 2^0 in row 1=a(0)
row 2: 2 3 . . . <- 2^1 in row 2=a(1)
row 3: 4 5 6 . . . <- 2^2 in row 3=a(2)
row 4: 7 8 9 10 . . <- 2^3 in row 4=a(3)
row 5: 11 12 13 14 15
row 6: 16 17 18 19 20 21 <- 2^4 in row 6=a(4).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Programs

Magma

[Round(Sqrt(2)^(n+1)): n in [0..50]]; // Vincenzo Librandi, Mar 24 2013

Mathematica

Table[Round[Sqrt[2]^(n+1)], {n, 0, 50}] (* Vincenzo Librandi, Mar 24 2013 *)

PARI

A057048(n)=round(sqrt(2^(n+1)))  /* for large values, an implementation using integer arithmetic would be preferable */ \\ M. F. Hasler, Feb 20 2012

PARI

a(n)=sqrtint(2^(n+1)) \\ Charles R Greathouse IV, Aug 19 2016

Python

from math import isqrt
def A057048(n): return -isqrt(m:=1<Chai Wah Wu, Jun 18 2024

Formula

a(2n-1) = 2^n, n > 0. - M. F. Hasler, Feb 20 2012