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A299533 Solution (b(n)) of the system of 3 complementary equations in Comments.

%I A299533 #8 May 05 2018 04:18:15
%S A299533 3,5,10,13,15,18,20,23,27,30,34,38,40,43,47,50,53,55,59,62,64,68,70,
%T A299533 73,77,79,82,86,89,92,93,99,101,104,108,111,114,115,120,122,126,129,
%U A299533 132,135,140,142,144,146,150,153,157,160,163,165,169,174,176,178
%N A299533 Solution (b(n)) of the system of 3 complementary equations in Comments.
%C A299533 Define sequences a(n), b(n), c(n) recursively:
%C A299533 a(n) = least new;
%C A299533 b(n) = least new > = a(n) + n + 1;
%C A299533 c(n) = a(n) + b(n);
%C A299533 where "least new k" means the least positive integer not yet placed.
%C A299533 ***
%C A299533 The sequences a,b,c partition the positive integers.
%C A299533 ***
%C A299533 Let x = be the greatest solution of 1/x + 1/(x+1) + 1/(2x+1) = 1. Then
%C A299533 x = 1/3 + (2/3)*sqrt(7)*cos((1/3)*arctan((3*sqrt(111))/67))
%C A299533 x = 2.07816258732933084676..., and a(n)/n - > x, b(n)/n -> x+1, and c(n)/n - > 2x+1. (The same limits occur in A298868 and A297838.)
%H A299533 Clark Kimberling, <a href="/A299533/b299533.txt">Table of n, a(n) for n = 0..1000</a>
%e A299533 n:   0   1   2   3   4   5   6   7   8   9  10
%e A299533 a:   1   2   6   8   9  11  12  14  17  19  22
%e A299533 b:   3   5  10  13  15  18  20  23  27  30  34
%e A299533 c:   4   7  16  21  24  29  32  37  44  49  56
%t A299533 z = 200;
%t A299533 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A299533 a = {}; b = {}; c = {}; n = 0;
%t A299533 Do[{n++;
%t A299533    AppendTo[a,
%t A299533     mex[Flatten[{a, b, c}], If[Length[a] == 0, 1, Last[a]]]],
%t A299533    AppendTo[b, mex[Flatten[{a, b, c}], Last[a] + n + 1]],
%t A299533    AppendTo[c, Last[a] + Last[b]]}, {z}];
%t A299533 (* _Peter J. C. Moses_, Apr 23 2018 *)
%t A299533 Take[a, 100] (* A297469 *)
%t A299533 Take[b, 100] (* A299533 *)
%t A299533 Take[c, 100] (* A299423 *)
%t A299533 (* _Peter J. C. Moses_, Apr 23 2018 *)
%Y A299533 Cf. A299634, A298868, A297838, A297469, A299423.
%K A299533 nonn,easy
%O A299533 0,1
%A A299533 _Clark Kimberling_, May 01 2018