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A326661 Rectangular array in 3 columns that solve the complementary equation c(n) = a(n) + b(3n), where a(1) = 1; see Comments.

%I A326661 #14 Feb 06 2023 10:01:36
%S A326661 1,2,7,3,4,16,5,6,25,8,9,35,10,11,43,12,13,52,14,15,61,17,18,71,19,20,
%T A326661 79,21,22,88,23,24,97,26,27,107,28,29,115,30,31,124,32,33,133,34,36,
%U A326661 142,37,38,151,39,40,160,41,42,169,44,45,179,46,47,187,48
%N A326661 Rectangular array in 3 columns that solve the complementary equation c(n) = a(n) + b(3n), where a(1) = 1; see Comments.
%C A326661 Let A = (a(n)), B = (b(n)), and C = (c(n)).  A unique solution (A,B,C) exists for the following conditions: (1) A,B,C must partition the positive integers, and (2) A and B are defined by mex (minimal excludant, as in A067017); that is, a(n) is the least "new" positive integer, and likewise for b(n).
%e A326661 c(1) = a(1) + b(3) >= 1 + 6, so that b(1) = mex{1} = 2; a(2) = mex{1,2} = 3; b(2) = mex{1,2,3} = 4; a(3)= mex{1,2,3,4} = 5, a(4) = mex{1,2,3,4,5} = 6, c(1) = 7.
%e A326661 n           a(n)      b(n)     c(n)
%e A326661 ---------------------------
%e A326661 1             1        2        7
%e A326661 2             3        4       16
%e A326661 3             5        6       25
%e A326661 4             8        9       35
%e A326661 5            10       11       43
%e A326661 6            12       13       52
%e A326661 7            14       15       61
%e A326661 8            17       18       74
%e A326661 9            19       20       79
%e A326661 10           21       22       88
%t A326661 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A326661 a = b = c = {}; h = 1; k = 3;
%t A326661 Do[Do[AppendTo[a,
%t A326661   mex[Flatten[{a, b, c}], Max[Last[a /. {} -> {0}], 1]]];
%t A326661   AppendTo[b, mex[Flatten[{a, b, c}], Max[Last[b /. {} -> {0}], 1]]], {k}];
%t A326661   AppendTo[c, a[[h Length[a]/k]] + Last[b]], {150}];
%t A326661 {a, b, c} // ColumnForm
%t A326661 a = Take[a, Length[c]]; b = Take[b, Length[c]];
%t A326661 Flatten[Transpose[{a, b, c}]](* _Peter J. C. Moses_, Jul 04 2019 *)
%Y A326661 Cf. A309157, A326662, A326664.
%K A326661 nonn,tabf,easy
%O A326661 1,2
%A A326661 _Clark Kimberling_, Jul 16 2019
%E A326661 Replaced a(0)->a(1) in NAME. - _R. J. Mathar_, Jun 19 2021