# [seqfan] Missing 5?

franktaw at netscape.net franktaw at netscape.net
Thu Sep 17 08:50:10 CEST 2009

```Consider the sequence (which I'll probably add soon), a(n) is the least
positive integer such that in the sequences a(n), a(n)+n, and a(n)+2n
no number occurs more than once.  (Note that if the a(n)+2n is left
out, this definition gives us the lower Wythoff sequence, A000201.)
This sequence starts:

1, 4, 7, 11, 12, 14, 9, 21, 18, 24, 28, 30, 25, 31, 32, 33, 35, 40, ...

Now, using the PARI program:

al(n)={local(u,r);
u=vector(5*n);r=vector(n);
for(i=1,n,
for(k=1,3*i,
if(!u[k]&&!u[k+i]&&!u[k+2*i],r[i]=k;u[k]=u[k+i]=u[k+2*i]=1;break)));
r}

I compute that a(1000) = 2211; and every integer between 1 and 2211 is
in one of the three specified sequences -- except for 5.

So ... does 5 eventually occur?  (It would have to be in the sequence
itself.)  Is there any other positive integer that does not occur?

Secondary question: is the sequence asymptotic to c*n, where c is the
root of (1/x + 1/(x+1) + 1/(x+2)) = 1?  (This reduces algebraically to
x^3 - 4x - 2 = 0; numerically it is 2.2143197433775351874154977....)