A164888 Smallest positive integer for each n such that the sequences a(n), a(n)+n, and a(n)+2n have no repeated terms.

1, 4, 7, 11, 12, 14, 9, 21, 18, 24, 28, 30, 25, 31, 32, 33, 35, 40, 41, 43, 46, 48, 55, 56, 57, 61, 64, 53, 66, 68, 71, 73, 75, 72, 77, 74, 84, 85, 86, 89, 90, 93, 96, 99, 100, 103, 97, 111, 114, 115, 116, 119, 120, 94, 126, 122, 117, 127, 130, 132, 136, 138, 142, 150, 134

Offset

1,2

Comments

If the a(n)+2n is left out, this definition gives us the lower Wythoff sequence, A000201.

It appears that the three sequences include every positive integer except 5.

It appears that the sequence is asymptotic to c*n, where c = 2.2143... is the positive root of x^3-4x-2. It appears that a(n) = c*n + O(log(n)); possibly even a(n) = c*n + O(1). (This polynomial is obtained by solving 1/x+1/(x+1)+1/(x+2) = 1.)

Links

Table of n, a(n) for n=1..65.

Example

The first term is 1; the sequences start 1; 2; 3. The smallest possible value for a(2) is then 4, giving 1,4; 2,6; and 3,8. a(3) cannot be 5, because a(3)+3 = 8 in the second sequence would then duplicate the 8 in the third sequence. a(3) = 7 works; the sequences to that point are 1,4,7; 2,6,10; 3,8,13.

Prog

(PARI) 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}

Crossrefs

Cf. A000201, A164889, A005228, A000027.

Sequence in context: A032547 A075630 A274341 * A023985 A023979 A319280

Adjacent sequences: A164885 A164886 A164887 * A164889 A164890 A164891

Keyword

nonn

Author

Franklin T. Adams-Watters, Sep 21 2009

Status

approved