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

1,2

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

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)

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

Crossrefs

Cf. A007814.

Sequence in context: A176506 A200596 A088422 * A097293 A296656 A303945

Adjacent sequences: A100830 A100831 A100832 * A100834 A100835 A100836

Keyword

nonn

Author

John W. Layman, Jan 07 2005

Status

approved