A090892 Solutions x to equation floor(x*r*floor(x/r)) = floor(x/r*floor(x*r)) when r = sqrt(2).

0, 1, 2, 3, 6, 9, 10, 12, 13, 16, 17, 19, 20, 23, 26, 27, 30, 33, 34, 36, 37, 40, 43, 44, 47, 50, 51, 53, 54, 57, 58, 60, 61, 64, 67, 68, 70, 71, 74, 75, 77, 78, 81, 84, 85, 88, 91, 92, 94, 95, 98, 99, 101, 102, 105, 108, 109, 111, 112, 115, 116, 118, 119, 122, 125, 126

Offset: 0

Author

Benoit Cloitre, Feb 15 2004

Keywords

• nonn

Comments

Terms >= 2 give numbers n satisfying: floor(sqrt(2)*n) is even. - Benoit Cloitre, May 27 2004

Essentially equivalent to A120752, see Fried link. - Charles R Greathouse IV, Jan 20 2023

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Sela Fried, Equivalent conditions for the nth element of the Beatty sequence B_sqrt(2) being even, arXiv preprint arXiv:2301.00644 [math.HO], 2022.

Crossrefs

Cf. A001951, A002193, A120752.

Programs

Mathematica

With[{r = Sqrt[2]}, Select[Range[0, 150], Floor[#*r*Floor[#/r]] == Floor[(#/r)*Floor[#*r]] &]] (* G. C. Greubel, Feb 06 2019 *)

PARI

r=sqrt(2); for(n=0,150, if(floor(n*r*floor(n/r))==floor(n/r*floor(n*r)), print1(n, ", "))) \\ G. C. Greubel, Feb 06 2019

Formula

It seems that a(n) = 2*n + o(n); conjecture : a(n) = 2*n + O(1).