cp's OEIS Frontend

A190431 a(n) = [(b*n+c)*r] - b*[n*r] - [c*r], where (r,b,c)=(golden ratio,3,1) and []=floor.

%I A190431 #12 Sep 08 2022 08:45:57
%S A190431 2,1,3,2,0,2,1,3,2,1,3,1,0,2,1,3,2,0,2,1,3,2,1,3,1,0,2,1,3,2,1,2,1,0,
%T A190431 2,1,3,2,0,2,1,3,2,1,3,1,0,2,1,3,2,1,2,1,3,2,1,3,2,0,2,1,3,2,1,2,1,0,
%U A190431 2,1,3,2,0,2,1,3,2,1,3,1,0,2,1,3,2,1,2,1,0,2,1,3,2,0,2,1,3,2,1,3,1,0,2,1,3,2,1,2,1,3,2,1,3,1,0,2,1,3,2,1,2,1,0,2,1,3,2,0,2,1,3,2,1,3,1,0
%N A190431 a(n) = [(b*n+c)*r] - b*[n*r] - [c*r], where (r,b,c)=(golden ratio,3,1) and []=floor.
%C A190431 Write a(n) = [(b*n+c)*r] - b*[n*r] - [c*r].  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 A190431 Examples:
%C A190431 (golden ratio,2,0):  A078588, A005653, A005652
%C A190431 (golden ratio,2,1):  A190427 - A190430
%C A190431 (golden ratio,3,0):  A140397 - A190400
%C A190431 (golden ratio,3,1):  A140431 - A190435
%C A190431 (golden ratio,3,2):  A140436 - A190439
%H A190431 G. C. Greubel, <a href="/A190431/b190431.txt">Table of n, a(n) for n = 1..10000</a>
%F A190431 a(n) = floor((3*n+1)*(1+sqrt(5))/2) - 3*floor(n*(1+sqrt(5))/2) - 1. - _G. C. Greubel_, Apr 06 2018
%t A190431 r = GoldenRatio; b = 3; c = 1;
%t A190431 f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
%t A190431 t = Table[f[n], {n, 1, 320}] (* A190431 *)
%t A190431 Flatten[Position[t, 0]] (* A190432 *)
%t A190431 Flatten[Position[t, 1]] (* A190433 *)
%t A190431 Flatten[Position[t, 2]] (* A190434 *)
%t A190431 Flatten[Position[t, 3]] (* A190435 *)
%o A190431 (PARI) for(n=1,100, print1(floor((3*n+1)*(1+sqrt(5))/2) - 3*floor(n*(1+sqrt(5))/2) - 1, ", ")) \\ _G. C. Greubel_, Apr 06 2018
%o A190431 (Magma) [Floor((3*n+1)*(1+Sqrt(5))/2) - 3*Floor(n*(1+Sqrt(5))/2) - 1: n in [1..100]]; // _G. C. Greubel_, Apr 06 2018
%Y A190431 Cf. A190432, A190433, A190434, A190435.
%K A190431 nonn
%O A190431 1,1
%A A190431 _Clark Kimberling_, May 10 2011