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A190544 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(2),4,0) and []=floor.

%I A190544 #14 Jul 03 2017 22:37:18
%S A190544 1,3,0,2,0,1,3,1,2,0,2,3,1,3,0,2,0,1,3,1,2,0,2,3,1,3,0,2,0,1,3,1,2,0,
%T A190544 1,3,1,2,0,2,3,1,3,0,2,0,1,3,1,2,0,2,3,1,3,0,2,0,1,3,1,2,0,2,3,1,3,0,
%U A190544 2,3,1,3,0,2,0,1,3,1,2,0,2,3,1,3,0,2,0,1,3,1,2,0,2,3,1,3,0,2,0,1,3,0,2,0,1,3,1,2,0,2,3,1,3,0,2,0,1,3,1,2,0,2,3,1,3,0,2,0
%N A190544 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(2),4,0) and []=floor.
%C A190544 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 A190544 Examples:
%C A190544 (golden ratio,2,1):  A190427-A190430
%C A190544 (sqrt(2),2,0):  A190480
%C A190544 (sqrt(2),2,1):  A190483-A190486
%C A190544 (sqrt(2),3,0):  A190487-A190490
%C A190544 (sqrt(2),3,1):  A190491-A190495
%C A190544 (sqrt(2),3,2):  A190496-A190500
%C A190544 (sqrt(2),4,c):  A190544-A190566
%H A190544 G. C. Greubel, <a href="/A190544/b190544.txt">Table of n, a(n) for n = 1..1000</a>
%F A190544 a(n) = [4nr] - 4*[nr], where r=sqrt(2).
%t A190544 r = Sqrt[2]; b = 4; c = 0;
%t A190544 f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
%t A190544 t = Table[f[n], {n, 1, 200}] (* A190544 *)
%t A190544 Flatten[Position[t, 0]]      (* A190545 *)
%t A190544 Flatten[Position[t, 1]]      (* A190546 *)
%t A190544 Flatten[Position[t, 2]]      (* A190547 *)
%t A190544 Flatten[Position[t, 3]]      (* A190548 *)
%Y A190544 Cf. A190545, A190546, A190547, A190548.
%K A190544 nonn
%O A190544 1,2
%A A190544 _Clark Kimberling_, May 12 2011