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%I #4 Apr 24 2018 12:10:52
%S 2,5,14,32,53,77,104,134,170,209,254,302,353,407,464,524,587,653,722,
%T 794,869,950,1034,1121,1211,1304,1403,1505,1610,1718,1829,1943,2060,
%U 2180,2303,2429,2558,2690,2825,2966,3110,3257,3407,3560,3716,3878,4043,4211
%N Solution (b(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(0) = 4:
%C a(n) = least new;
%C b(n) = a(n-1)+c(n-1);
%C c(n) = 2 a(n) + 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="/A296502/b296502.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 3 6 7 8 9 10 12 13 15
%e b: 2 5 14 32 53 77 104 134 170 209
%e c: 4 11 26 46 69 95 124 158 196 239
%t z = 300;
%t mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t a = {1}; b = {2}; c = {4}; n = 1;
%t Do[{n++, AppendTo[a, mex[Flatten[{a, b, c}], 1]],
%t AppendTo[b, a[[n - 1]] + c[[n - 1]]],
%t AppendTo[c, 2 Last[a] + Last[b]]}, {z}];
%t Take[a, 100] (* A296484 *)
%t Take[b, 100] (* A296502 *)
%t Take[c, 100] (* A297149 *)
%Y Cf. A299634, A296484, A297149.
%K nonn,easy
%O 0,1
%A _Clark Kimberling_, Apr 24 2018
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