A190483 - OEIS

%I #20 Mar 12 2021 07:53:30

%S 1,2,0,1,0,1,2,1,1,0,1,2,1,2,0,1,0,1,2,0,1,0,1,2,1,1,0,1,0,1,2,0,1,0,

%T 1,2,1,1,0,1,2,1,2,0,1,0,1,2,1,1,0,1,2,1,1,0,1,0,1,2,0,1,0,1,2,1,1,0,

%U 1,2,1,2,0,1,0,1,2,1,1,0,1,2,1,2,0,1,0,1,2,0,1,0,1,2,1,1,0,1,0,1,2,0,1,0,1,2,1,1,0,1,2,1,2,0

%N a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(2),2,1) 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,2,1):  A190427-A190430

%C (sqrt(2),2,0):  A190480

%C (sqrt(2),2,1):  A190483-A190486

%C (sqrt(2),3,0):  A190487-A190490

%C (sqrt(2),3,1):  A190491-A190495

%C (sqrt(2),3,2):  A190496-A190500

%H G. C. Greubel, <a href="/A190483/b190483.txt">Table of n, a(n) for n = 1..1000</a>

%t r = Sqrt[2]; b = 2; c = 1;

%t f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];

%t t = Table[f[n], {n, 1, 200}]  (* A190483 *)

%t Flatten[Position[t, 0]]   (* A190484 *)

%t Flatten[Position[t, 1]]   (* A190485 *)

%t Flatten[Position[t, 2]]   (* A190486 *)

%o (Python)

%o from sympy import sqrt, floor

%o r=sqrt(2)

%o def a(n): return floor((2*n + 1)*r) - 2*floor(n*r) - floor(r)

%o print([a(n) for n in range(1, 501)]) # _Indranil Ghosh_, Jul 02 2017

%Y Cf. A190484, A190485, A190486.

%K nonn

%O 1,2

%A _Clark Kimberling_, May 11 2011