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%I #4 Apr 18 2018 19:49:25
%S 1,4,5,7,8,9,10,12,13,14,15,17,19,20,21,22,23,24,25,28,29,30,31,32,33,
%T 34,36,37,38,39,40,41,42,44,46,47,48,49,50,51,52,53,54,55,56,58,59,61,
%U 62,63,64,65,66,67,68,69,71,72,73,74,75,76,77,78,80,81
%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 k >= a(n) + b(n-1);
%C c(n) = a(n) + 2 b(n);
%C where "least new k" means the least positive integer not yet placed. The sequences a,b,c partition the positive integers.
%H Clark Kimberling, <a href="/A298871/b298871.txt">Table of n, a(n) for n = 0..1000</a>
%e n: 0 1 2 3 4 5 6 7 8 9
%e a: 1 4 5 7 8 9 10 12 13 14
%e b: 2 6 11 18 26 35 45 57 70 84
%e c: 3 16 27 43 60 30 79 100 126 153
%t z = 400;
%t mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t a = {1}; b = {2}; c = {3};
%t Do[{AppendTo[a, mex[Flatten[{a, b, c}], 1]],
%t AppendTo[b, mex[Flatten[{a, b, c}], Last[a] + Last[b]]],
%t AppendTo[c, Last[a] + 2 Last[b]]}, {z}];
%t Take[a, 100] (* A298871 *)
%t Take[b, 100] (* A298872 *)
%t Take[c, 100] (* A298873 *)
%Y Cf. A299634, A298872, A298873, A298874.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Apr 18 2018
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