A006346 The Sally sequence: the length of repetition avoided in A006345.

0, 1, 1, 2, 1, 3, 1, 1, 3, 2, 1, 6, 3, 2, 1, 3, 1, 1, 6, 3, 2, 4, 1, 1, 3, 2, 1, 3, 1, 6, 4, 2, 1, 2, 4, 3, 1, 8, 3, 2, 1, 6, 3, 2, 1, 3, 1, 1, 6, 3, 2, 4, 1, 1, 3, 2, 1, 3, 1, 30, 6, 3, 2, 4, 1, 1, 3, 2, 1, 3, 1, 6, 4, 2, 1, 2, 4, 3, 1, 8, 3, 2, 1, 6, 3, 2, 1, 3, 1, 1, 6, 3, 2, 4, 1, 1, 3, 2, 1, 3, 1, 6, 4, 2

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy
• look

Comments

In the first 1000 terms, only patterns of lengths 0, 1, 2, 3, 4, 6, 8, 24, 30 and 108 are avoided. - T. D. Noe, Oct 14 2006

In the first 20000 terms, patterns of lengths 0, 1, 2, 3, 4, 5, 6, 7, 8, 16, 18, 24, 30, 108, 528, 552, 1298, 2752, 5876 are avoided. - Robert Israel, Jun 22 2015

In the first 10^6 terms, patterns of lengths 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 24, 30, 36, 60, 72, 108, 318, 372, 414, 420, 444, 522, 528, 546, 552, 1280, 1292, 1298, 1632, 2620, 2752, 5876, 6203, 6218, 13912, 14312, 17220, 17580, 31532, 87650 are avoided. - Ray Chandler, Jun 24 2015

References

• N. S. Hellerstein, personal communication to N. J. A. Sloane.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Robert Israel and Hugo van der Sanden, Table of n, a(n) for n = 1..50000 (1..1000 from T. D. Noe, 1001..20000 from Robert Israel)
• N. Hellerstein, Letter to N. J. A. Sloane, 1978
• N. Hellerstein, M. Gardner, & S. Kim, Correspondence related to the Linus and Sally sequences, 1977
• N. J. A. Sloane, Illustration of initial terms
• Eric Weisstein's World of Mathematics, Sally Sequence.

Crossrefs

Cf. A006345.

Programs

Maple

LDS:= proc(L)
  local Cands, r, m;
  Cands:= {$1..floor(nops(L)/2)};
  r:= 0;
  for m from 1 while nops(Cands) > 0 do
     Cands:= select(c -> L[-m] = L[-c-m], Cands);
     if min(Cands) = m then
        r:= m;
        Cands:= subs(m=NULL,Cands);
     fi
   od;
   r
end proc:
A:= 1: B:= 0;
for n from 2 to 1000 do
  a1:= LDS([A,1]);
  a2:= LDS([A,2]);
  if a1 < a2 then A:= A,1; B:= B, a2 else A:= A,2; B:= B, a1 fi;
od:
seq(B[i],i=1..1000); # Robert Israel, Jun 22 2015

Mathematica

max = 120;
LDS[L_] := Module[{Cands, r, m},
  Cands = Range[Floor[Length[L]/2]];
  r = 0;
  For[m = 1, Length[Cands] > 0, m++,
    Cands = Select[Cands, L[[-m]] == L[[-#-m]]&];
    If[Min[Cands] == m, r = m;
    Cands = ReplaceAll[Cands, m -> Nothing]]
  ];
r];
A = {1}; B = {0};
For[n = 2, n <= max, n++,
  a1 = LDS[Append[A, 1]];
  a2 = LDS[Append[A, 2]];
  If[a1 < a2,
    AppendTo[A, 1]; AppendTo[B, a2],
    AppendTo[A, 2]; AppendTo[B, a1]
  ]
];
Table[B[[i]], {i, 1, max}] (* Jean-François Alcover, Jun 08 2018, after Robert Israel's Maple program *)

Extensions

More terms from Naohiro Nomoto, May 21 2001