A190436 a(n) = [(b*n+c)*r] - b*[n*r] - [c*r], where (r,b,c)=(golden ratio,3,2) and []=floor.

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

Offset: 1

Author

Clark Kimberling, May 10 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,2,0): 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

(golden ratio,4,c): A140440 - A190461

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A190437, A190438, A190439, A190440.

Programs

Mathematica

r = GoldenRatio; b = 3; c = 2;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 320}]
Flatten[Position[t, 0]] (* A190437 *)
Flatten[Position[t, 1]] (* A190438 *)
Flatten[Position[t, 2]] (* A190439 *)
Flatten[Position[t, 3]] (* A302253 *)