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A298868 Solution (a(n)) of the system of 3 complementary equations in Comments.

%I A298868 #8 Apr 18 2018 02:59:25
%S A298868 1,4,6,8,11,14,15,17,19,21,24,26,27,29,32,33,34,37,41,42,45,46,48,52,
%T A298868 53,54,57,58,59,61,64,67,70,72,73,74,77,79,82,83,87,90,92,93,94,96,98,
%U A298868 100,101,104,105,107,111,113,115,118,119,120,122,125,126,127
%N A298868 Solution (a(n)) of the system of 3 complementary equations in Comments.
%C A298868 Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:
%C A298868 a(n) = least new;
%C A298868 b(n) = least new k >= a(n) + n;
%C A298868 c(n) = a(n) + b(n);
%C A298868 where "least new k" means the least positive integer not yet placed.
%C A298868 ***
%C A298868 The sequences a,b,c partition the positive integers. Let x = be the greatest solution of 1/x + 1/(x+1) + 1/(2x+1) = 1. Then
%C A298868 x =  1/3 + (2/3)*sqrt(7)*cos((1/3)*arctan((3*sqrt(111))/67));
%C A298868 x = 2.07816258732933084676..., and a(n)/n -> x, b(n)/n -> x+1, and c(n)/n -> 2x+1.
%H A298868 Clark Kimberling, <a href="/A298868/b298868.txt">Table of n, a(n) for n = 0..1000</a>
%e A298868 n:   0    1    2    3    4    5    6    7    8    9
%e A298868 a:   1    4    6    8   11   14   15   17   19   21
%e A298868 b:   2    5    7   10   12   16   20   22   25   28
%e A298868 c:   3    9   13   18   23   30   35   39   44   49
%t A298868 z = 400;
%t A298868 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A298868 a = {1}; b = {2}; c = {}; AppendTo[c, Last[a] + Last[b]]; n = 0;
%t A298868 Do[{n++, AppendTo[a, mex[Flatten[{a, b, c}], 1]],
%t A298868    AppendTo[b, mex[Flatten[{a, b, c}], a[[n]] + n]],
%t A298868    AppendTo[c, Last[a] + Last[b]]}, {z}];
%t A298868 Take[a, 100] (* A298868 *)
%t A298868 Take[b, 100] (* A298869 *)
%t A298868 Take[c, 100] (* A298870 *)
%t A298868 (* _Peter J. C. Moses_, Apr 08 2018 *)
%Y A298868 Cf. A299634, A298869, A298870.
%K A298868 nonn,easy
%O A298868 0,2
%A A298868 _Clark Kimberling_, Apr 17 2018