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A190561 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(2),4,3) and []=floor.

%I A190561 #10 Jul 04 2017 18:28:12
%S A190561 1,3,1,2,0,2,3,1,3,0,2,4,1,3,1,2,0,2,3,1,3,0,2,4,1,3,0,2,0,1,3,1,2,0,
%T A190561 2,3,1,3,0,2,4,1,3,1,2,0,2,3,1,3,0,2,4,1,3,1,2,0,1,3,1,2,0,2,3,1,3,0,
%U A190561 2,4,1,3,1,2,0,2,3,1,3,0,2,4,1,3,1,2,0,2,3,1,3,0,2,3,1,3,0,2,0,1,3,1,2,0,2,3,1,3,0,2,4,1,3,1,2,0,2,3,1,3,0,2,4,1,3,1,2,0,1,3,1,2,0,2,3,1,3,0
%N A190561 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(2),4,3) and []=floor.
%C A190561 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 A190561 Examples:
%C A190561 (golden ratio,2,1):  A190427-A190430
%C A190561 (sqrt(2),2,0):  A190480-A190482
%C A190561 (sqrt(2),2,1):  A190483-A190486
%C A190561 (sqrt(2),3,0):  A190487-A190490
%C A190561 (sqrt(2),3,1):  A190491-A190495
%C A190561 (sqrt(2),3,2):  A190496-A190500
%C A190561 (sqrt(2),4,c):  A190544-A190566
%H A190561 G. C. Greubel, <a href="/A190561/b190561.txt">Table of n, a(n) for n = 1..1000</a>
%t A190561 r = Sqrt[2]; b = 4; c = 3;
%t A190561 f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
%t A190561 t = Table[f[n], {n, 1, 200}] (* A190561 *)
%t A190561 Flatten[Position[t, 0]]          (* A190562 *)
%t A190561 Flatten[Position[t, 1]]          (* A190563 *)
%t A190561 Flatten[Position[t, 2]]          (* A190564 *)
%t A190561 Flatten[Position[t, 3]]          (* A190565 *)
%t A190561 Flatten[Position[t, 4]]          (* A190566 *)
%Y A190561 Cf. A190562 to A190566.
%K A190561 nonn
%O A190561 1,2
%A A190561 _Clark Kimberling_, May 12 2011