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A090892
Solutions x to equation floor(x*r*floor(x/r)) = floor(x/r*floor(x*r)) when r = sqrt(2).
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
0,3
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
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.
FORMULA
It seems that a(n) = 2*n + o(n); conjecture : a(n) = 2*n + O(1).
MATHEMATICA
With[{r = Sqrt[2]}, Select[Range[0, 150], Floor[#*r*Floor[#/r]] == Floor[(#/r)*Floor[#*r]] &]] (* G. C. Greubel, Feb 06 2019 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Feb 15 2004
STATUS
approved