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

%I #4 Apr 24 2018 12:10:59

%S 4,11,26,46,69,95,124,158,196,239,286,336,389,445,504,566,631,699,770,

%T 844,923,1006,1092,1181,1273,1370,1471,1575,1682,1792,1905,2021,2140,

%U 2262,2387,2515,2646,2780,2919,3062,3208,3357,3509,3664,3824,3988,4155,4325

%N Solution (c(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="/A297149/b297149.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, A296502.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, Apr 24 2018