Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #14 Apr 09 2018 02:56:18
%S 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 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 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 a(n) = [(b*n+c)*r] - b*[n*r] - [c*r], where (r,b,c)=(golden ratio,3,2) and []=floor.
%C 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 Examples:
%C (golden ratio,2,0): A078588, A005653, A005652
%C (golden ratio,2,1): A190427 - A190430
%C (golden ratio,3,0): A140397 - A190400
%C (golden ratio,3,1): A140431 - A190435
%C (golden ratio,3,2): A140436 - A190439
%C (golden ratio,4,c): A140440 - A190461
%H G. C. Greubel, <a href="/A190436/b190436.txt">Table of n, a(n) for n = 1..10000</a>
%t r = GoldenRatio; b = 3; c = 2;
%t f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
%t t = Table[f[n], {n, 1, 320}]
%t Flatten[Position[t, 0]] (* A190437 *)
%t Flatten[Position[t, 1]] (* A190438 *)
%t Flatten[Position[t, 2]] (* A190439 *)
%t Flatten[Position[t, 3]] (* A302253 *)
%Y Cf. A190437, A190438, A190439, A190440.
%K nonn
%O 1,1
%A _Clark Kimberling_, May 10 2011