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A304814 Solution (b(n)) of the complementary equation a(n) = b(3n) + b(5n); see Comments. 3

%I

%S 1,3,4,5,6,7,8,9,10,11,13,14,15,16,17,18,19,20,22,23,24,25,26,27,28,

%T 30,31,32,33,34,35,36,37,38,40,41,42,43,44,45,46,47,49,50,51,52,53,54,

%U 55,56,58,59,60,61,62,63,64,65,67,68,69,70,71,72,73,75,76

%N Solution (b(n)) of the complementary equation a(n) = b(3n) + b(5n); see Comments.

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

%C b(n) = least new,

%C a(n) = b(3n) + b(5n),

%C where "least new" means the least positive integer not yet placed. Empirically, {a(n) - 8*n: n >= 0} = {2,3} and {7*b(n) - 8*n: n >= 0} = {7,8,9,10,11,12,13,14,15}. See A304799 for a guide to related sequences.

%H Clark Kimberling, <a href="/A304814/b304814.txt">Table of n, a(n) for n = 0..9999</a>

%e b(0) = 1, so that a(0) = 2. Since a(1) = b(3) + b(5), we must have a(1) >= 10, so that b(1) = 3, b(2) = 4, b(3) = 5, b(4) = 6, b(5) = 7, and a(1) = 12.

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

%t h = 3; k = 5; 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] (* A304813 *)

%t Take[b, 200] (* A304814 *)

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

%Y Cf. A304799, A304813.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, May 30 2018

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Last modified June 14 00:35 EDT 2021. Contains 345016 sequences. (Running on oeis4.)