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%I #20 Sep 08 2022 08:45:57
%S 1,0,0,1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,
%T 1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,
%U 1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,1,1
%N a(n) = [(b*n+c)*r] - b*[n*r] - [c*r], where (r,b,c)=(sqrt(3),2,0) and [ ] = floor.
%C Write a(n) = [(b*n+c)*r] - b*[n*r] - [c*r]. If r>0 and b and c are integers satisfying b >= 2 and 0 <= c <= b-1, then 0 <= a(n) <= b. The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b. These b+1 position sequences comprise a partition of the positive integers.
%C Examples:
%C (golden ratio,2,1): A190427-A190430
%C (sqrt(2),2,0): A197879, A120243, A120749
%C (sqrt(2),2,1): A190483-A190486
%C (sqrt(2),3,0): A190487-A190490
%C (sqrt(2),3,1): A190491-A190495
%C (sqrt(2),3,2): A190496-A190500
%C (sqrt(2),4,c): A190544-A190566
%C (sqrt(3),2,0): A190669-A190671
%H G. C. Greubel, <a href="/A190669/b190669.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = floor(2*n*sqrt(3)) - 2*floor(n*sqrt(3)).
%t r = Sqrt[3]; b = 2; c = 0;
%t f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
%t t = Table[f[n], {n, 1, 200}] (* A190669 *)
%t Flatten[Position[t, 0]] (* A190670 *)
%t Flatten[Position[t, 1]] (* A190671 *)
%o (PARI) for(n=1,100, print1(floor(2*n*sqrt(3)) - 2*floor(n*sqrt(3)), ", ")) \\ _G. C. Greubel_, Apr 20 2018
%o (Magma) [Floor(2*n*Sqrt(3)) - 2*Floor(n*Sqrt(3)): n in [1..100]]; // _G. C. Greubel_, Apr 20 2018
%Y Cf. A190670, A190671.
%K nonn
%O 1
%A _Clark Kimberling_, May 16 2011
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