A190451 - OEIS

%I #8 Mar 30 2012 18:57:28

%S 2,1,3,2,0,3,1,4,2,0,3,1,0,2,1,3,2,0,3,1,4,2,1,3,2,0,2,1,3,2,0,3,1,0,

%T 2,1,3,2,0,3,1,4,2,1,3,1,0,2,1,3,2,0,3,1,4,2,1,3,2,0,3,1,3,2,0,3,1,0,

%U 2,1,3,2,0,3,1,4,2,1,3,2,0,2,1,3,2,0,3,1,0,2,1,3,2,0,3,1,4,2,0,3,1,0,2,1,3,2,0,3,1,4,2,1,3,2,0,3,1,3,2,0,3,1,0

%N [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,2) 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

%t r = GoldenRatio; b = 4; 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]]

%t Flatten[Position[t, 1]]

%t Flatten[Position[t, 2]]

%t Flatten[Position[t, 3]]

%t Flatten[Position[t, 4]]

%Y Cf. A190428, A190453-A190455.

%K nonn

%O 1,1

%A _Clark Kimberling_, May 10 2011