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A298741 Solution (a(n)) of the complementary equation in Comments. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified February 24 18:45 EST 2021. Contains 341579 sequences. (Running on oeis4.)