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A190436 a(n) = [(b*n+c)*r] - b*[n*r] - [c*r], where (r,b,c)=(golden ratio,3,2) and []=floor. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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

LINKS

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

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 *)

CROSSREFS

Cf. A190437, A190438, A190439, A190440.

Sequence in context: A255195 A194317 A096810 * A133696 A195050 A127371

Adjacent sequences:  A190433 A190434 A190435 * A190437 A190438 A190439

KEYWORD

nonn

AUTHOR

Clark Kimberling, May 10 2011

STATUS

approved

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Last modified June 2 10:43 EDT 2020. Contains 334770 sequences. (Running on oeis4.)