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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.

%I A190436 #14 Apr 09 2018 02:56:18
%S A190436 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,
%T A190436 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,
%U A190436 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
%N A190436 a(n) = [(b*n+c)*r] - b*[n*r] - [c*r], where (r,b,c)=(golden ratio,3,2) and []=floor.
%C A190436 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.
%C A190436 Examples:
%C A190436 (golden ratio,2,0):  A078588, A005653, A005652
%C A190436 (golden ratio,2,1):  A190427 - A190430
%C A190436 (golden ratio,3,0):  A140397 - A190400
%C A190436 (golden ratio,3,1):  A140431 - A190435
%C A190436 (golden ratio,3,2):  A140436 - A190439
%C A190436 (golden ratio,4,c):  A140440 - A190461
%H A190436 G. C. Greubel, <a href="/A190436/b190436.txt">Table of n, a(n) for n = 1..10000</a>
%t A190436 r = GoldenRatio; b = 3; c = 2;
%t A190436 f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
%t A190436 t = Table[f[n], {n, 1, 320}]
%t A190436 Flatten[Position[t, 0]] (* A190437 *)
%t A190436 Flatten[Position[t, 1]] (* A190438 *)
%t A190436 Flatten[Position[t, 2]] (* A190439 *)
%t A190436 Flatten[Position[t, 3]] (* A302253 *)
%Y A190436 Cf. A190437, A190438, A190439, A190440.
%K A190436 nonn
%O A190436 1,1
%A A190436 _Clark Kimberling_, May 10 2011