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A298870 Solution (c(n)) of the system of 3 complementary equations in Comments.

%I A298870 #8 May 05 2018 04:18:46
%S A298870 3,9,13,18,23,30,35,39,44,49,55,62,65,69,75,80,84,88,97,102,108,112,
%T A298870 116,123,129,132,138,143,145,150,155,162,169,175,179,183,187,193,199,
%U A298870 204,211,218,225,228,231,235,240,246,249,255,259,263,270,277,282,288
%N A298870 Solution (c(n)) of the system of 3 complementary equations in Comments.
%C A298870 Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:
%C A298870 a(n) = least new;
%C A298870 b(n) = least new k >= a(n) + n;
%C A298870 c(n) = a(n) + b(n);
%C A298870 where "least new k" means the least positive integer not yet placed.
%C A298870 ***
%C A298870 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 A298870 x = 1/3 + (2/3)*sqrt(7)*cos((1/3)*arctan((3*sqrt(111))/67))
%C A298870 x = 2.07816258732933084676..., and a(n)/n - > x, b(n)/n -> x+1, and c(n)/n - > 2x+1.
%H A298870 Clark Kimberling, <a href="/A298870/b298870.txt">Table of n, a(n) for n = 0..1000</a>
%e A298870 n:   0    1    2    3    4    5    6    7    8    9
%e A298870 a:   1    4    6    8   11   14   15   17   19   21
%e A298870 b:   2    5    7   10   12   16   20   22   25   28
%e A298870 c:   3    9   13   18   23   30   35   39   44   49
%t A298870 z = 400;
%t A298870 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A298870 a = {1}; b = {2}; c = {}; AppendTo[c, Last[a] + Last[b]]; n = 0;
%t A298870 Do[{n++, AppendTo[a, mex[Flatten[{a, b, c}], 1]],
%t A298870    AppendTo[b, mex[Flatten[{a, b, c}], a[[n]] + n]],
%t A298870    AppendTo[c, Last[a] + Last[b]]}, {z}];
%t A298870 Take[a, 100] (* A298868 *)
%t A298870 Take[b, 100] (* A298869 *)
%t A298870 Take[c, 100] (* A298870 *)
%t A298870 (* _Peter J. C. Moses_, Apr 08 2018 *)
%Y A298870 Cf. A299634, A298868, A298869.
%K A298870 nonn,easy
%O A298870 0,1
%A A298870 _Clark Kimberling_, Apr 18 2018