A288382 - OEIS

%I #13 Feb 18 2021 02:02:24

%S 1,2,3,5,8,13,22,39,72,137,266,523,1036,2061,4110,8207,16400,32785,

%T 65554,131091,262164,524309,1048598,2097175,4194328,8388633,16777242

%N Positions of 0 in A288381; complement of A288383.

%C a(n+1)/a(n)-> 2.

%C Appears to be the same as A052968 (apart from the offset). - _R. J. Mathar_, Jun 14 2017

%C This conjecture is proved in A288381. - _Michel Dekking_, Feb 18 2021

%F a(n) = -1 + A288133(n-1) for n >= 2.

%F Conjectures from _Colin Barker_, Jun 10 2017: (Start)

%F G.f.: x*(1 - 2*x + x^3 - x^4) / ((1 - x)^2*(1 - 2*x)).

%F a(n) = -1 + 2^(n-3) + n for n>2.

%F a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3) for n>5.

%F (End)

%F Barker's conjectures are implied by Mathar's conjecture. - _Michel Dekking_, Feb 18 2021

%t s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];

%t w[n_] := StringReplace[w[n - 1], {"00" -> "0001", "1" -> "11"}]

%t Table[w[n], {n, 0, 8}]

%t st = ToCharacterCode[w[11]] - 48   (* A288381 *)

%t Flatten[Position[st, 0]]  (* A288382 *)

%t Flatten[Position[st, 1]]  (* A288383 *)

%Y Cf. A288381, A288383.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Jun 10 2017