A298173 Solution (a(n)) of the complementary equation in Comments.

1, 4, 16, 50, 155, 468, 1410, 4234, 12709, 38132, 114404, 343218, 1029663, 3088997, 9267001, 27801012, 83403047, 250209151, 750627465, 2251882406, 6755647231, 20266941705, 60800825129, 182402475400, 547207426215, 1641622278659, 4924866835993, 14774600507994

Offset

0,2

Comments

Define sequences a(n) and b(n) recursively, starting with a(0) = 1, b(0) = 2:

b(n) = least new;

a(n) = 3*a(n-1) + x(0)*b(n) + x(1)*b(n-1) + ... + x(n)b(0),

where "least new k" means the least positive integer not yet placed, and x(k) = (-1)^k for k >= 0.

***

It appears that a(n)/a(n-1) -> 3 and that {a(n) - 3*a(n-1), n >= 1} is unbounded.

Links

Table of n, a(n) for n=0..27.

Example

b(1) = least not in {a(0),b(0)} = 3;
a(1) = 3*a(0) + b(1) - b(0) = 3*1 + 3 - 2 = 4.

Mathematica

mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
c = 3; a = {1}; b = {2}; x = {-1};
Do[AppendTo[b, mex[Flatten[{a, b}], 1]];
 AppendTo[x, -Last[x]];
 AppendTo[a, c Last[a] - (Reverse[x] b // Total)], {25}]
Grid[{Join[{"n"}, Range[0, # - 1]], Join[{"a(n)"}, Take[a, #]],
    Join[{"b(n)"}, Take[b, #]], Join[{"x(n)"}, Take[-x, #]]},
   Alignment -> ".",
   Dividers -> {{2 -> Red, -1 -> Blue}, {2 -> Red, -1 -> Blue}}] &[10]
(* Peter J. C. Moses, May 10 2018 *)

Crossrefs

Cf. A298741, A298877.

Sequence in context: A227675 A345325 A203094 * A323932 A121184 A203840

Adjacent sequences: A298170 A298171 A298172 * A298174 A298175 A298176

Keyword

nonn,easy

Author

Clark Kimberling, May 12 2018

Status

approved