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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
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.)
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
KEYWORD
nonn
AUTHOR
STATUS
approved