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A304811 Solution (a(n)) of the complementary equation a(n) = b(2n) + b(5n) ; see Comments. 3

%I

%S 2,11,19,26,34,43,51,58,67,75,83,91,99,107,114,123,131,138,146,155,

%T 163,170,179,187,195,203,211,219,226,235,243,250,258,267,275,282,291,

%U 299,306,314,323,331,338,347,355,363,370,379,387,394,403,411,418,426

%N Solution (a(n)) of the complementary equation a(n) = b(2n) + b(5n) ; see Comments.

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

%C b(n) = least new,

%C a(n) = b(2n) + 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} = {6,7,8,9,10,11,12,13}. See A304799 for a guide to related sequences.

%H Clark Kimberling, <a href="/A304811/b304811.txt">Table of n, a(n) for n = 0..9999</a>

%e b(0) = 1, so that a(0) = 2. Since a(1) = b(2) + b(5), we must have a(1) >= 9, so that b(1) = 3, b(2) = 4, b(3) = 5, b(4) = 6, b(5) = 7, b(6) = 8, and a(1) = 11.

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

%t h = 2; 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] (* A304811 *)

%t Take[b, 200] (* A304812 *)

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

%Y Cf. A304799, A304812.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, May 30 2018

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Last modified June 23 13:01 EDT 2021. Contains 345401 sequences. (Running on oeis4.)