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

0,1

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

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 *)

Prog

(PARI) for(n=0, 100, print1(floor(floor(n*sqrt(2)^2) - sqrt(2)*floor(n*sqrt(2))), ", ")) \\ G. C. Greubel, Jan 27 2018
(Magma) [Floor(Floor(n*Sqrt(2)^2) - Sqrt(2)*Floor(n*Sqrt(2))): n in [0..100]]; // 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

Crossrefs

Cf. A007069. Positions of ones: A059649.

Cf. A002193 (sqrt(2)).

Sequence in context: A188472 A286044 A129272 * A288707 A079261 A354028

Adjacent sequences: A059645 A059646 A059647 * A059649 A059650 A059651

Keyword

nonn

Author

Antti Karttunen, Feb 03 2001

Status

approved