A190698 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(3),4,1) and [ ]=floor.

3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 0, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 4, 3, 1, 0, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 4, 2, 1, 0, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 0, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 4, 3, 1, 0, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 4, 2, 1, 0, 3, 2, 1, 4, 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, 4, 3, 2, 1, 4, 3, 2, 1, 4, 3, 1, 0, 3, 2, 1, 4

Offset

1,1

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Mathematica

r = Sqrt[3]; b = 4; c = 1;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 200}] (* A190698 *)
Flatten[Position[t, 0]]      (* A190699 *)
Flatten[Position[t, 1]]      (* A190700 *)
Flatten[Position[t, 2]]      (* A190701 *)
Flatten[Position[t, 3]]      (* A190702 *)
Flatten[Position[t, 4]]      (* A190703 *)

Crossrefs

Cf. A190699-A190703.

Sequence in context: A224381 A190704 A346611 * A283183 A327467 A347832

Adjacent sequences: A190695 A190696 A190697 * A190699 A190700 A190701

Keyword

nonn

Author

Clark Kimberling, May 17 2011

Status

approved