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A299423 Solution (c(n)) of the system of 3 complementary equations in Comments.

%I A299423 #8 May 05 2018 04:18:11
%S A299423 4,7,16,21,24,29,32,37,44,49,56,63,66,71,78,83,88,91,98,103,106,113,
%T A299423 116,121,128,131,136,143,147,152,154,164,168,173,180,185,189,191,200,
%U A299423 203,210,214,219,225,234,237,240,243,250,255,262,267,272,275,281,291
%N A299423 Solution (c(n)) of the system of 3 complementary equations in Comments.
%C A299423 Define sequences a(n), b(n), c(n) recursively:
%C A299423 a(n) = least new;
%C A299423 b(n) = least new > = a(n) + n + 1;
%C A299423 c(n) = a(n) + b(n);
%C A299423 where "least new k" means the least positive integer not yet placed.
%C A299423 ***
%C A299423 The sequences a,b,c partition the positive integers.
%C A299423 ***
%C A299423 Let x = be the greatest solution of 1/x + 1/(x+1) + 1/(2x+1) = 1. Then
%C A299423 x = 1/3 + (2/3)*sqrt(7)*cos((1/3)*arctan((3*sqrt(111))/67))
%C A299423 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 A299423 Clark Kimberling, <a href="/A299423/b299423.txt">Table of n, a(n) for n = 0..1000</a>
%e A299423 n:   0   1   2   3   4   5   6   7   8   9  10
%e A299423 a:   1   2   6   8   9  11  12  14  17  19  22
%e A299423 b:   3   5  10  13  15  18  20  23  27  30  34
%e A299423 c:   4   7  16  21  24  29  32  37  44  49  56
%t A299423 z = 200;
%t A299423 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A299423 a = {}; b = {}; c = {}; n = 0;
%t A299423 Do[{n++;
%t A299423    AppendTo[a,
%t A299423     mex[Flatten[{a, b, c}], If[Length[a] == 0, 1, Last[a]]]],
%t A299423    AppendTo[b, mex[Flatten[{a, b, c}], Last[a] + n + 1]],
%t A299423    AppendTo[c, Last[a] + Last[b]]}, {z}];
%t A299423 (* _Peter J. C. Moses_, Apr 23 2018 *)
%t A299423 Take[a, 100] (* A297469 *)
%t A299423 Take[b, 100] (* A299533 *)
%t A299423 Take[c, 100] (* A299423 *)
%t A299423 (* _Peter J. C. Moses_, Apr 23 2018 *)
%Y A299423 Cf. A299634, A298868, A297838, A297469, A299533.
%K A299423 nonn,easy
%O A299423 0,1
%A A299423 _Clark Kimberling_, May 01 2018