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

%I #5 Mar 30 2012 18:57:11

%S 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 141,46,15,7,7165,5075,1671,590,186,53,18,8,30349,21494,7072,2492,778,

%U 214,64,20,9,128558,91046,29951,10549,3286,896,259,71,23,10,544578

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

%C (row 1)=-1+A049652.

%C (column 1)=A000027.

%C (column 2)=A001950=(u(n)), or simply u.

%C (column 3)=u(u(n))+l(l(n)), or simply uu+ll.

%C (column 4)=u(uu+ll)+l(ul+lu),

%C whereas Column 4 of the lower Beatty array

%C is u(ul+lu)+l(uu+ll).

%C U(n,k)-L(n,k)=n for n>=1, k>=0.

%F Here we introduce Beatty arrays. Suppose that

%F ((u(1),u(2),...) and (l(1),l(2),...) are the Beatty

%F sequences of positive real numbers r and s=r/(1-r), where

%F r<s. For n>=1, let

%F U(n,0)=n, U(n,1)=u(1), L(n,0)=0, L(n,1)=l(1),

%F and for k>=2 let x=floor(r*u(k-1)), y=floor(r*l(k-1)),

%F a=x+u(k-1), b=x, c=y+l(k-1), d=y,

%F U(n,k)=a+d, L(n,k)=b+c. We call U and L the upper and

%F lower Beatty arrays of r (and of s). Note that

%F U(n,k)-L(n,k)=U(n,1)-L(n,1) for all n>=1 and k>=1.

%e Northwest corner of the array:

%e 1.....2.....6....23....95....400...

%e 2.....5....17....68...284...1199...

%e 3.....7....24....95...396...1671...

%e 4....10....35...141...590...2492...

%Y Cf. A181886, A000201, A001950, A000045.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Nov 18 2010

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