A190440 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,0) and []=floor.

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

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,2,0): A078588, A005653, A005652

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

(golden ratio,3,0): A140397-A190400

(golden ratio,3,1): A140431-A190435

(golden ratio,3,2): A140436-A190439

Links

Table of n, a(n) for n=1..112.

Formula

a(n)=[4nr]-4[nr], where r=golden ratio.

Mathematica

r = GoldenRatio;
f[n_] := Floor[4*n*r] - 4*Floor[n*r];
t = Table[f[n], {n, 1, 320}] (* A190440 *)
Flatten[Position[t, 0]]  (* A190240 *)
Flatten[Position[t, 1]]  (* A190249 *)
Flatten[Position[t, 2]]  (* A190442 *)
Flatten[Position[t, 3]]  (* A190443 *)
Flatten[Position[t, 4]]  (* A190248 *)

Crossrefs

Cf. A190889, A190442, A190443, A190251.

Sequence in context: A158853 A238762 A269517 * A054372 A070679 A177443

Adjacent sequences: A190437 A190438 A190439 * A190441 A190442 A190443

Keyword

nonn

Author

Clark Kimberling, May 10 2011

Status

approved