A138330 Beatty discrepancy (defined in A138253) giving the closeness of the pair (A136497,A136498) to the Beatty pair (A001951,A001952).

1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1

Offset

1,2

Comments

Old definition was "Beatty discrepancy of the complementary equation b(n) = a(a(n)) + a(n)".

Links

Muniru A Asiru, Table of n, a(n) for n = 1..1000

Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.

Formula

a(n) = d(n) - c(c(n)) - c(n), where c(n) = A001951 and d(n) = A001952.

a(n) = 2*n - A007069(n). - Benoit Cloitre, May 08 2008

a(n) = A059648(n+1) + 1. - Michel Dekking, Nov 11 2018

Example

d(1) - c(c(1)) - c(1) =  3 - 1 - 1 = 1;
d(2) - c(c(2)) - c(2) =  6 - 2 - 2 = 2;
d(3) - c(c(3)) - c(3) = 10 - 5 - 4 = 1;
d(4) - c(c(4)) - c(4) = 13 - 7 - 5 = 1.

Maple

a:=n->2*n-floor(sqrt(2)*floor(sqrt(2)*n)): seq(a(n), n=1..120); # Muniru A Asiru, Nov 11 2018

Mathematica

Table[2 n - Floor[Sqrt[2] Floor[Sqrt[2] n]], {n, 1, 100}] (* Vincenzo Librandi, Nov 12 2018 *)

Prog

(PARI) a(n)=2*n-floor(sqrt(2)*floor(sqrt(2)*n)) \\ Benoit Cloitre, May 08 2008
(Magma) [2*n - Floor(Sqrt(2)*Floor(Sqrt(2)*n)): n in [1..100]]; // Vincenzo Librandi, Nov 12 2018
(Python)
from math import isqrt
def A138330(n): return (m:=n<<1)-isqrt(isqrt(n*m)**2<<1) # Chai Wah Wu, Aug 29 2022

Crossrefs

Cf. A001951, A001952, A136497, A136498, A138253, A059648.

Sequence in context: A115721 A279497 A359305 * A128591 A354747 A254444

Adjacent sequences: A138327 A138328 A138329 * A138331 A138332 A138333

Keyword

nonn

Author

Clark Kimberling, Mar 14 2008

Extensions

Definition revised by N. J. A. Sloane, Dec 16 2018

Status

approved