A190693 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(3),4,0) and [ ]=floor.

2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 3, 2, 0, 3, 2, 1, 0, 3

Offset: 1

Author

Clark Kimberling, May 17 2011

Keywords

• nonn

Comments

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.

Examples:

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

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

(sqrt(2),2,1): A190483-A190486

(sqrt(2),3,0): A190487-A190490

(sqrt(2),3,1): A190491-A190495

(sqrt(2),3,2): A190496-A190500

(sqrt(2),4,c): A190544-A190566

Crossrefs

Cf. A190694-A190697.

Programs

Mathematica

r = Sqrt[3]; b = 4; c = 0;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 200}] (* A190693 *)
Flatten[Position[t, 0]]      (* A190694 *)
Flatten[Position[t, 1]]      (* A190695 *)
Flatten[Position[t, 2]]      (* A190696 *)
Flatten[Position[t, 3]]      (* A190697 *)

Formula

a(n)=[4n*sqrt(3)]-4[n*sqrt(3)].