

A181661


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


5



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
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OFFSET

1,2


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.


LINKS

Table of n, a(n) for n=1..56.


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/(1r), where
r<s. For n>=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(k1)), y=floor(r*l(k1)),
a=x+u(k1), b=x, c=y+l(k1), 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.


EXAMPLE

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.
Sequence in context: A209773 A209767 A122070 * A144160 A275142 A200226
Adjacent sequences: A181658 A181659 A181660 * A181662 A181663 A181664


KEYWORD

nonn,tabl


AUTHOR

Clark Kimberling, Nov 18 2010


STATUS

approved



