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A190704 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(3),4,2) and [ ]=floor.

%I A190704 #10 Mar 18 2023 09:05:07
%S A190704 3,2,1,4,3,2,0,3,2,1,0,3,2,1,4,3,2,1,4,3,1,0,3,2,1,0,3,2,1,4,3,2,1,4,
%T A190704 2,1,0,3,2,1,0,3,2,1,4,3,2,1,3,2,1,0,3,2,1,4,3,2,1,4,3,2,0,3,2,1,0,3,
%U A190704 2,1,4,3,2,1,4,3,1,0,3,2,1,0,3,2,1,4,3,2,1,4,2,1,0,3,2,1,0,3,2,1,4,3,2,0,3,2,1,0,3,2,1,4,3,2,1,4,3,1,0,3,2,1,0,3,2,1,4,3,2,1,4,2
%N A190704 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(3),4,2) and [ ]=floor.
%C A190704 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 A190704 Examples:
%C A190704 (golden ratio,2,1):  A190427-A190430
%C A190704 (sqrt(2),2,0):  A190480-A190482
%C A190704 (sqrt(2),2,1):  A190483-A190486
%C A190704 (sqrt(2),3,0):  A190487-A190490
%C A190704 (sqrt(2),3,1):  A190491-A190495
%C A190704 (sqrt(2),3,2):  A190496-A190500
%C A190704 (sqrt(2),4,c):  A190544-A190566
%t A190704 r = Sqrt[3]; b = 4; c = 2;
%t A190704 f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
%t A190704 t = Table[f[n], {n, 1, 200}] (* A190704 *)
%t A190704 Flatten[Position[t, 0]]      (* A190673 *)
%t A190704 Flatten[Position[t, 1]]      (* A190706 *)
%t A190704 Flatten[Position[t, 2]]      (* A190707 *)
%t A190704 Flatten[Position[t, 3]]      (* A190708 *)
%t A190704 Flatten[Position[t, 4]]      (* A190709 *)
%t A190704 With[{r=Sqrt[3],nn=140},Table[Floor[(4n+2)r]-4Floor[n r]-Floor[2r],{n,nn}]] (* _Harvey P. Dale_, Mar 18 2023 *)
%Y A190704 Cf. A190673, A190706-A190709.
%K A190704 nonn
%O A190704 1,1
%A A190704 _Clark Kimberling_, May 17 2011