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A298173 Solution (a(n)) of the complementary equation in Comments. 2
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 (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) = 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: A217949 A227675 A203094 * A323932 A121184 A203840

Adjacent sequences:  A298170 A298171 A298172 * A298174 A298175 A298176

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, May 12 2018

STATUS

approved

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Last modified May 16 15:55 EDT 2021. Contains 343949 sequences. (Running on oeis4.)