A190688 - OEIS

%I #6 Mar 30 2012 18:57:28

%S 2,1,1,3,2,1,0,3,2,1,0,2,2,1,3,2,1,0,3,2,1,0,2,2,1,0,2,1,1,3,2,1,0,3,

%T 2,1,0,2,2,1,0,2,1,1,3,2,1,0,3,2,1,0,2,2,1,3,2,1,1,3,2,1,0,3,2,1,0,2,

%U 1,1,3,2,1,0,3,2,1,0,2,2,1,0,2,1,1,3,2,1,0,3,2,1,0,2,2,1,0,2,1,1,3,2,1,0,3,2,1,0,2,2,1,3,2,1,1,3,2,1,0,3,2,1,0,2,1,1,3,2,1,0,3,2

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

%C 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 Examples:

%C (golden ratio,2,1):  A190427-A190430

%C (sqrt(2),2,0):  A190480-A190482

%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

%t r = Sqrt[3]; b = 3; c = 2;

%t f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];

%t t = Table[f[n], {n, 1, 200}] (* A190688 *)

%t Flatten[Position[t, 0]]      (* A190689 *)

%t Flatten[Position[t, 1]]      (* A190690 *)

%t Flatten[Position[t, 2]]      (* A190691 *)

%t Flatten[Position[t, 3]]      (* A190692 *)

%Y Cf. A190689-A190692.

%K nonn

%O 1,1

%A _Clark Kimberling_, May 17 2011