cp's OEIS Frontend

A190457 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,3) and []=floor.

%I A190457 #11 Mar 30 2025 10:57:45
%S A190457 3,1,4,2,1,3,2,4,3,1,4,2,0,3,1,4,2,1,3,2,4,3,1,4,2,1,3,2,4,3,1,3,2,0,
%T A190457 3,1,4,2,1,3,2,4,3,1,4,2,1,3,1,4,2,1,3,2,4,3,1,4,2,1,3,2,4,3,1,4,2,0,
%U A190457 3,1,4,2,1,3,2,4,3,1,4,2,1,3,2,4,2,1,3,2,0,3,1,4,2,1,3,2,4,3,1,4,2,1,3,1,4,2,1,3,2,4,3,1,4,2,1,3,2,4,3,1,3,2,0,3,1,4,2,1,3,2,4,3
%N A190457 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,3) and []=floor.
%C A190457 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 A190457 Examples:
%C A190457 (golden ratio,2,0):  A078588, A005653, A005652
%C A190457 (golden ratio,2,1):  A190427-A190430
%C A190457 (golden ratio,3,0):  A140397-A190400
%C A190457 (golden ratio,3,1):  A140431-A190435
%C A190457 (golden ratio,3,2):  A140436-A190439
%C A190457 (golden ratio,4,c):  A190440-A190461
%H A190457 Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, <a href="https://arxiv.org/abs/2503.19696">Fibonacci-like partitions and their associated piecewise-defined permutations</a>, arXiv:2503.19696 [math.CO], 2025. See p. 4.
%t A190457 r = GoldenRatio; b = 4; c = 3;
%t A190457 f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
%t A190457 t = Table[f[n], {n, 1, 320}]
%t A190457 Flatten[Position[t, 0]]
%t A190457 Flatten[Position[t, 1]]
%t A190457 Flatten[Position[t, 2]]
%t A190457 Flatten[Position[t, 3]]
%t A190457 Flatten[Position[t, 4]]
%Y A190457 Cf. A190458-A190461 and A190463.
%K A190457 nonn
%O A190457 1,1
%A A190457 _Clark Kimberling_, May 10 2011