A059648 a(n) = [[(k^2)*n]-(k*[k*n])], where k = sqrt(2) and [] is the floor function.

0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 0

Author

Antti Karttunen, Feb 03 2001

Keywords

• nonn

Comments

The values of (floor((k^2)*j)-(k*(floor(k*j)))) for j=0..20, with k=sqrt(2), are 0, 0.585786, 1.171572, 0.343144, 0.928930, 0.100502, 0.68629, 1.27207, 0.44365, 1.02943, 0.20100, 0.78679, 1.37258, 0.54415, 1.12993, 0.30151, 0.88729, 0.05886, 0.64465, 1.23044, 0.40201

Links

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

Crossrefs

Cf. A007069. Positions of ones: A059649.

Cf. A002193 (sqrt(2)).

Programs

Magma

[Floor(Floor(n*Sqrt(2)^2) - Sqrt(2)*Floor(n*Sqrt(2))): n in [0..100]]; // G. C. Greubel, Jan 27 2018

Maple

Digits := 89; floor_diffs_floored(sqrt(2),120); floor_diffs_floored := proc(k,upto_n) local j; [seq(floor(floor((k^2)*j)-(k*(floor(k*j)))),j=0..upto_n)]; end;

Mathematica

With[{k = Sqrt[2]}, Table[Floor[Floor[k^2*j] - k*Floor[k*j]], {j, 0, 104}]] (* Jean-François Alcover, Mar 06 2016 *)

PARI

for(n=0, 100, print1(floor(floor(n*sqrt(2)^2) - sqrt(2)*floor(n*sqrt(2))), ", ")) \\ G. C. Greubel, Jan 27 2018

Python

from math import isqrt
def A059648(n): return (m:=n<<1)-1-isqrt(isqrt(n*m)**2<<1) if n else 0 # Chai Wah Wu, Aug 29 2022