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A304807 Solution (a(n)) of the complementary equation a(n) = b(2n) + b(3n) ; see Comments. 3

%I

%S 2,9,14,20,27,32,39,44,51,57,62,69,75,81,87,92,99,105,111,117,122,129,

%T 134,140,147,152,159,164,170,177,182,189,194,200,207,212,219,225,231,

%U 237,242,249,255,260,267,272,279,285,290,297,302,309,314,320,327

%N Solution (a(n)) of the complementary equation a(n) = b(2n) + b(3n) ; see Comments.

%C Define complementary sequences a(n) and b(n) recursively:

%C b(n) = least new,

%C a(n) = b(2n) + b(3n),

%C where "least new" means the least positive integer not yet placed. Empirically, {a(n) - 6*n: n >= 0} = {2,3} and {5*b(n) - 6*n: n >= 0} = {4,5,6,7,8,9}. See A304799 for a guide to related sequences.

%H Clark Kimberling, <a href="/A304807/b304807.txt">Table of n, a(n) for n = 0..10000</a>

%e b(0) = 1, so that a(0) = 2. Since a(1) = b(2) + b(3), we must have a(1) >= 7, so that b(1) = 3, b(2) = 4, b(3) = 5, b(4) = 6, and a(1) = 9.

%t mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);

%t h = 2; k = 3; a = {}; b = {1};

%t AppendTo[a, mex[Flatten[{a, b}], 1]];

%t Do[Do[AppendTo[b, mex[Flatten[{a, b}], Last[b]]], {k}];

%t AppendTo[a, Last[b] + b[[1 + (Length[b] - 1)/k h]]], {500}];

%t Take[a, 200] (* A304807 *)

%t Take[b, 200] (* A304808 *)

%t (* _Peter J. C. Moses_, May 14 2008 *)

%Y Cf. A304799, A304808.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, May 28 2018

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Last modified June 23 02:24 EDT 2021. Contains 345395 sequences. (Running on oeis4.)