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

1, 2, 2, 6, 5, 3, 23, 17, 7, 4, 95, 68, 24, 10, 5, 400, 284, 95, 35, 13, 6, 1692, 1199, 396, 141, 46, 15, 7, 7165, 5075, 1671, 590, 186, 53, 18, 8, 30349, 21494, 7072, 2492, 778, 214, 64, 20, 9, 128558, 91046, 29951, 10549, 3286, 896, 259, 71, 23, 10, 544578

Offset: 1

Author

Clark Kimberling, Nov 18 2010

Keywords

• nonn
• tabl

Comments

(row 1)=-1+A049652.

(column 1)=A000027.

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

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

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

whereas Column 4 of the lower Beatty array

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

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

Examples

			Northwest corner of the array:
  1     2     6    23    95    400 ...
  2     5    17    68   284   1199 ...
  3     7    24    95   396   1671 ...
  4    10    35   141   590   2492 ...
		

Crossrefs

Cf. A181886, A000201, A001950, A000045.

Formula

Here we introduce Beatty arrays. Suppose that

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

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

r=1, let

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

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

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

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

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

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