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A297468 Solution (b(n)) of the system of 2 complementary equations in Comments.

%I A297468 #4 Apr 25 2018 08:33:11
%S A297468 3,4,5,6,7,8,9,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,
%T A297468 29,30,32,33,34,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,
%U A297468 54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71
%N A297468 Solution (b(n)) of the system of 2 complementary equations in Comments.
%C A297468 Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, a(1) = 1, b(0) = 3; for n >= 1,
%C A297468 a(2n) = 3*a(n) + b(n);
%C A297468 a(2n+1) = 3*a(n-1) + n;
%C A297468 b(n) = least new;
%C A297468 where "least new k" means the least positive integer not yet placed.  The sequences (a(n)) and (b(n)) are complementary.
%H A297468 Clark Kimberling, <a href="/A297468/b297468.txt">Table of n, a(n) for n = 0..1000</a>
%e A297468 n:   0  1   2   3   4   5   6   7   8
%e A297468 a:   1  2  10  31  35  95  99 108 112
%e A297468 b:   3  4   5   6   7   8   9  11  12
%t A297468 z = 300;
%t A297468 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A297468 a = {1, 2}; b = {3};
%t A297468 Do[AppendTo[b, mex[Flatten[{a, b}], Last[b]]];
%t A297468  AppendTo[a, 3 a[[#/2 + 1]] + b[[#/2 + 1]]] &[Length[a]];
%t A297468  AppendTo[a, 3 a[[(# + 3)/2]] + (# - 1)/2] &[Length[a]], {z}]
%t A297468 Take[a, 100]  (* A297467 *)
%t A297468 Take[b, 100]  (* A297468 *)
%t A297468 (* _Peter J. C. Moses_,  Apr 22 2018 *)
%Y A297468 Cf. A299634, A297467.
%K A297468 nonn,easy
%O A297468 0,1
%A A297468 _Clark Kimberling_, Apr 24 2018