A049473 Nearest integer to n/sqrt(2).

0, 1, 1, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 11, 11, 12, 13, 13, 14, 15, 16, 16, 17, 18, 18, 19, 20, 21, 21, 22, 23, 23, 24, 25, 25, 26, 27, 28, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 35, 36, 37, 37, 38, 39, 40, 40, 41, 42, 42, 43, 44, 45, 45, 46, 47, 47

Offset

0,4

Comments

a(n) = floor(n*sqrt(2)) - floor(n/sqrt(2)). Indeed, the equation {(nearest integer to n/r) = floor(nr) - floor(n/r) for all n>=0} has exactly two solutions: sqrt(2) and -sqrt(2). - Clark Kimberling, Dec 18 2003

Let s(n) = zeta(3) - Sum_{k=1..n} 1/k^3. Conjecture: for n >=1, s(a(n)) < 1/n^2 < s(a(n)-1), and the difference sequence of A049473 consists solely of 0's and 1, in positions given by the nonhomogeneous Beatty sequences A001954 and A001953, respectively. - Clark Kimberling, Oct 05 2014

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

Mathematica

Round[Range[0, 70]/Sqrt[2]] (* Harvey P. Dale, Feb 17 2015 *)

Prog

(PARI) a(n)=round(n/sqrt(2)) \\ Charles R Greathouse IV, Sep 02 2015
(Magma) [0] cat [Round(n/Sqrt(2)): n in [1..100]]; // G. C. Greubel, Jan 27 2018

Crossrefs

Cf. A091087.

Sequence in context: A189730 A249569 A094500 * A154951 A095769 A080820

Adjacent sequences: A049470 A049471 A049472 * A049474 A049475 A049476

Keyword

nonn

Author

N. J. A. Sloane

Status

approved