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A190483 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(2),2,1) and []=floor.

%I A190483 #20 Mar 12 2021 07:53:30
%S A190483 1,2,0,1,0,1,2,1,1,0,1,2,1,2,0,1,0,1,2,0,1,0,1,2,1,1,0,1,0,1,2,0,1,0,
%T A190483 1,2,1,1,0,1,2,1,2,0,1,0,1,2,1,1,0,1,2,1,1,0,1,0,1,2,0,1,0,1,2,1,1,0,
%U A190483 1,2,1,2,0,1,0,1,2,1,1,0,1,2,1,2,0,1,0,1,2,0,1,0,1,2,1,1,0,1,0,1,2,0,1,0,1,2,1,1,0,1,2,1,2,0
%N A190483 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(2),2,1) and []=floor.
%C A190483 Write a(n)=[(bn+c)r]-b[nr]-[cr].  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 A190483 Examples:
%C A190483 (golden ratio,2,1):  A190427-A190430
%C A190483 (sqrt(2),2,0):  A190480
%C A190483 (sqrt(2),2,1):  A190483-A190486
%C A190483 (sqrt(2),3,0):  A190487-A190490
%C A190483 (sqrt(2),3,1):  A190491-A190495
%C A190483 (sqrt(2),3,2):  A190496-A190500
%H A190483 G. C. Greubel, <a href="/A190483/b190483.txt">Table of n, a(n) for n = 1..1000</a>
%t A190483 r = Sqrt[2]; b = 2; c = 1;
%t A190483 f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
%t A190483 t = Table[f[n], {n, 1, 200}]  (* A190483 *)
%t A190483 Flatten[Position[t, 0]]   (* A190484 *)
%t A190483 Flatten[Position[t, 1]]   (* A190485 *)
%t A190483 Flatten[Position[t, 2]]   (* A190486 *)
%o A190483 (Python)
%o A190483 from sympy import sqrt, floor
%o A190483 r=sqrt(2)
%o A190483 def a(n): return floor((2*n + 1)*r) - 2*floor(n*r) - floor(r)
%o A190483 print([a(n) for n in range(1, 501)]) # _Indranil Ghosh_, Jul 02 2017
%Y A190483 Cf. A190484, A190485, A190486.
%K A190483 nonn
%O A190483 1,2
%A A190483 _Clark Kimberling_, May 11 2011