A298741 Solution (a(n)) of the complementary equation in Comments.

1, 6, 19, 45, 102, 215, 445, 904, 1826, 3669, 7359, 14738, 29500, 59023, 118073, 236172, 472376, 944782, 1889599, 3779231, 7558500, 15117036, 30234113, 60468265, 120936574, 241873190, 483746427, 967492899, 1934985848, 3869971744, 7739943541, 15479887133

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) = 2*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, x(0) = 2, and x(k) = (-1)^k for k >= 1.

***

It appears that a(n)/a(n-1) -> 2 and that {a(n) - 2*a(n-1), n >=1 } is unbounded.

Links

Table of n, a(n) for n=0..31.

Example

b(1) = least not in {a(0),b(0)} = 3;
a(1) = 2*a(0) + 2 b(1) - b(0) = 2*1 +2*3 - 2 = 6.

Mathematica

mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
c = 2; a = {1}; b = {2}; x = {2};
Do[AppendTo[b, mex[Flatten[{a, b}], 1]];
 AppendTo[x, -Sign[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}}] &[18]
(* Peter J. C. Moses, May 10 2018 *)

Crossrefs

Cf. A298173, A298877.

Sequence in context: A299265 A005712 A299278 * A070893 A272047 A267829

Adjacent sequences: A298738 A298739 A298740 * A298742 A298743 A298744

Keyword

nonn,easy

Author

Clark Kimberling, May 12 2018

Status

approved