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A304809 Solution (a(n)) of the complementary equation a(n) = b(2n) + b(4n) ; see Comments. 3

%I

%S 2,10,17,23,31,38,44,52,59,65,73,80,86,94,101,107,115,122,128,136,143,

%T 149,157,164,170,178,185,191,199,206,212,220,227,233,241,248,254,262,

%U 269,275,283,290,296,304,311,317,325,332,338,346,353,359,367,374,380

%N Solution (a(n)) of the complementary equation a(n) = b(2n) + b(4n) ; see Comments.

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

%C b(n) = least new,

%C a(n) = b(2n) + b(4n),

%C where "least new" means the least positive integer not yet placed. Empirically, {a(n) - 7*n: n >= 0} = {2,3} and {6*b(n) - 7*n: n >= 0} = {5,6,7,8,9,10,11}. See A304799 for a guide to related sequences.

%H Clark Kimberling, <a href="/A304809/b304809.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(4), we must have a(1) >= 8, so that b(1) = 3, b(2) = 4, b(3) = 5, b(4) = 6, b(5) = 7, and a(1) = 10.

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

%t h = 2; k = 4; 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] (* A304809 *)

%t Take[b, 200] (* A304810 *)

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

%Y Cf. A304799, A304810.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, May 28 2018

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Last modified June 24 08:51 EDT 2021. Contains 345416 sequences. (Running on oeis4.)