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%I #6 Mar 30 2012 18:57:28
%S 3,1,4,2,0,3,1,4,2,1,3,2,0,3,1,4,2,1,3,2,4,3,1,3,2,0,3,1,4,2,1,3,2,0,
%T 3,1,4,2,1,3,1,4,2,1,3,2,0,3,1,4,2,1,3,2,4,3,1,4,2,0,3,1,4,2,1,3,2,0,
%U 3,1,4,2,1,3,2,4,2,1,3,2,0,3,1,4,2,1,3,2,0,3,1,4,2,0,3,1,4,2,1,3,2,0,3,1,4,2,1,3,2,4,3,1,3,2,0,3,1,4,2,1,3,2,0,3,1,4,2,1,3,1,4,2,1,3,2,0
%N [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,1) 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): A190440-A190461
%t r = GoldenRatio; b = 4; c = 1;
%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]]
%t Flatten[Position[t, 1]]
%t Flatten[Position[t, 2]]
%t Flatten[Position[t, 3]]
%t Flatten[Position[t, 4]]
%Y Cf. A190446-A190450, A190427.
%K nonn
%O 1,1
%A _Clark Kimberling_, May 10 2011
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