A006346 The Sally sequence: the length of repetition avoided in A006345. (Formerly M0126)

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

1,4

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.

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 *)

Crossrefs

Cf. A006345.

Sequence in context: A229214 A349685 A218578 * A244740 A088742 A256435

Adjacent sequences: A006343 A006344 A006345 * A006347 A006348 A006349

Keyword

nonn,nice,easy,look

Author

N. J. A. Sloane

Extensions

More terms from Naohiro Nomoto, May 21 2001

Status

approved