%I #11 May 01 2018 03:01:05
%S 1,4,5,7,9,12,15,16,17,20,21,25,27,28,29,33,34,35,36,39,45,46,47,48,
%T 52,56,57,58,60,61,62,64,65,67,74,75,76,78,79,80,81,87,88,94,95,97,
%U 100,102,103,104,105,106,107,108,110,114,117,123,124,125,126,127
%N Solution (a(n)) of the system of 3 complementary equations in Comments.
%C Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:
%C a(n) = least new;
%C b(n) = least new > = a(n) + n + 1;
%C c(n) = a(n) + b(n);
%C where "least new k" means the least positive integer not yet placed.
%C ***
%C The sequences a,b,c partition the positive integers.
%C ***
%C Let x = be the greatest solution of 1/x + 1/(x+1) + 1/(2x+1) = 1. Then
%C x = 1/3 + (2/3)*sqrt(7)*cos((1/3)*arctan((3*sqrt(111))/67))
%C x = 2.07816258732933084676..., and a(n)/n - > x, b(n)/n -> x+1, and c(n)/n - > 2x+1.
%C (The same limits occur in A298868 and A297469.)
%H Clark Kimberling, <a href="/A297838/b297838.txt">Table of n, a(n) for n = 0..1000</a>
%e n: 0 1 2 3 4 5 6 7 8 9 10
%e a: 1 4 5 7 9 12 15 16 17 20 21
%e b: 2 6 8 11 14 19 22 24 26 30 32
%e c: 3 10 13 18 23 31 37 40 43 50 53
%t z=200;
%t mex[list_,start_]:=(NestWhile[#+1&,start,MemberQ[list,#]&]);
%t a={1};b={2};c={3};n=0;
%t Do[{n++;
%t AppendTo[a,mex[Flatten[{a,b,c}],If[Length[a]==0,1,Last[a]]]],
%t AppendTo[b,mex[Flatten[{a,b,c}],Last[a]+n+1]],
%t AppendTo[c,Last[a]+Last[b]]},{z}];
%t Take[a,100] (* A297838 *)
%t Take[b,100] (* A298170 *)
%t Take[c,100] (* A298418 *)
%t (* _Peter J. C. Moses_, Apr 23 2018 *)
%Y Cf. A299634, A298868, A297469, A298170, A298418.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Apr 25 2018
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