A100833 Smallest positive palindrome-free and squarefree sequence.

1, 2, 3, 1, 2, 4, 1, 2, 3, 1, 2, 5, 1, 2, 3, 1, 2, 4, 1, 2, 3, 1, 2, 6, 1, 2, 3, 1, 2, 4, 1, 2, 3, 1, 2, 5, 1, 2, 3, 1, 2, 4, 1, 2, 3, 1, 2, 7, 1, 2, 3, 1, 2, 4, 1, 2, 3, 1, 2, 5, 1, 2, 3, 1, 2, 4, 1, 2, 3, 1, 2, 6, 1, 2, 3, 1, 2, 4, 1, 2, 3, 1, 2, 5, 1, 2, 3, 1, 2, 4, 1, 2, 3, 1, 2, 8, 1, 2, 3, 1, 2, 4, 1, 2, 3

Offset: 1

Author

John W. Layman, Jan 07 2005

Keywords

• nonn

Comments

a(1)=1 and, for n>1, a(n) is the smallest positive integer such that no subsequence of two or more consecutive terms is a palindrome and no subsequence of 2k consecutive terms, k>=1, consists of two copies of the same subsequence.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007814.

Programs

Maple

B[1]:= 1: B[2]:= 2:
for n from 3 to 200 do
  for c from 1 do
    if B[n-1]=c or B[n-2]=c then next fi;
    Cs:= ListTools:-Reverse(select(t -> B[t]=c, [$ceil(n/2)..n-3]));
    good:= true;
    for k in Cs do
      if andmap(t -> (B[k-t]=B[n-t]),[$1..n-k-1]) then good:= false; break fi
    od;
    if good then B[n]:= c; break fi;
  od;
od:
seq(B[i],i=1..200); # Robert Israel, Jan 17 2019

Formula

Conjectures from Robert Israel, Jan 17 2019: (Start)

a(n) = 1 if n == 1 (mod 3).

a(n) = 2 if n == 2 (mod 3).

Otherwise a(n) = 3 + A007814(n/3). (End)