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A181661 Upper Beatty array of the golden ratio, (1+sqrt(5))/2.

%I A181661 #8 Feb 10 2025 05:48:20
%S A181661 1,2,2,6,5,3,23,17,7,4,95,68,24,10,5,400,284,95,35,13,6,1692,1199,396,
%T A181661 141,46,15,7,7165,5075,1671,590,186,53,18,8,30349,21494,7072,2492,778,
%U A181661 214,64,20,9,128558,91046,29951,10549,3286,896,259,71,23,10,544578
%N A181661 Upper Beatty array of the golden ratio, (1+sqrt(5))/2.
%C A181661 (row 1)=-1+A049652.
%C A181661 (column 1)=A000027.
%C A181661 (column 2)=A001950=(u(n)), or simply u.
%C A181661 (column 3)=u(u(n))+l(l(n)), or simply uu+ll.
%C A181661 (column 4)=u(uu+ll)+l(ul+lu),
%C A181661 whereas Column 4 of the lower Beatty array
%C A181661 is u(ul+lu)+l(uu+ll).
%C A181661 U(n,k)-L(n,k)=n for n>=1, k>=0.
%F A181661 Here we introduce Beatty arrays.  Suppose that
%F A181661 (u(1),u(2),...) and (l(1),l(2),...) are the Beatty
%F A181661 sequences of positive real numbers r and s=r/(1-r), where
%F A181661 r<s.  For n>=1, let
%F A181661 U(n,0)=n, U(n,1)=u(1), L(n,0)=0, L(n,1)=l(1),
%F A181661 and for k>=2 let x=floor(r*u(k-1)), y=floor(r*l(k-1)),
%F A181661 a=x+u(k-1), b=x, c=y+l(k-1), d=y,
%F A181661 U(n,k)=a+d, L(n,k)=b+c.  We call U and L the upper and
%F A181661 lower Beatty arrays of r (and of s).  Note that
%F A181661 U(n,k)-L(n,k)=U(n,1)-L(n,1) for all n>=1 and k>=1.
%e A181661 Northwest corner of the array:
%e A181661   1     2     6    23    95    400 ...
%e A181661   2     5    17    68   284   1199 ...
%e A181661   3     7    24    95   396   1671 ...
%e A181661   4    10    35   141   590   2492 ...
%Y A181661 Cf. A181886, A000201, A001950, A000045.
%K A181661 nonn,tabl
%O A181661 1,2
%A A181661 _Clark Kimberling_, Nov 18 2010