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A309157 Rectangular array in 3 columns that solve the complementary equation c(n) = a(n) + b(2n), where a(1) = 1; see Comments.

%I A309157 #28 Feb 12 2021 16:03:17
%S A309157 1,2,5,3,4,12,6,7,20,8,9,26,10,11,33,13,14,41,15,16,47,17,18,54,19,21,
%T A309157 61,22,23,68,24,25,75,27,28,83,29,30,89,31,32,96,34,35,104,36,37,110,
%U A309157 38,39,117,40,42,124,43,44,131,45,46,138,48,49,146,50,51
%N A309157 Rectangular array in 3 columns that solve the complementary equation c(n) = a(n) + b(2n), where a(1) = 1; see Comments.
%C A309157 Let A = (a(n)), B = (b(n)), and C = (c(n)).  A unique solution (A,B,C) exists for these 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).
%H A309157 Clark Kimberling and Peter J. C. Moses, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Kimberling/kimb23.html">Complementary Equations with Advanced Subscripts</a>, J. Int. Seq. 24 (2021) Article 21.3.3.
%e A309157 c(1) = a(1) + b(2) > = 1 + 3, so that
%e A309157 a(2) = mex{1,2} = 3;
%e A309157 b(2) = mex{1,2,3} = 4;
%e A309157 c(1) = 5.
%e A309157 Then c(2) = a(2) + b(4) >= 3 + 8, so that
%e A309157 a(3) = 6, b(3) = 7;
%e A309157 a(4) = 8, b(4) = 9;
%e A309157 c(2) = a(2) + b(4) = 3 + 9 = 12.
%e A309157    n    a(n) b(n) c(n)
%e A309157   --------------------
%e A309157    1      1    2    5
%e A309157    2      3    4   12
%e A309157    3      6    7   20
%e A309157    4      8    9   26
%e A309157    5     10   11   33
%e A309157    6     13   14   41
%e A309157    7     15   16   47
%e A309157    8     17   18   54
%e A309157    9     19   21   61
%e A309157   10     22   23   68
%t A309157 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A309157 a = b = c = {}; h = 1; k = 2;
%t A309157 Do[Do[AppendTo[a,
%t A309157   mex[Flatten[{a, b, c}], Max[Last[a /. {} -> {0}], 1]]];
%t A309157   AppendTo[b, mex[Flatten[{a, b, c}], Max[Last[b /. {} -> {0}], 1]]], {k}];
%t A309157   AppendTo[c, a[[h Length[a]/k]] + Last[b]], {150}];
%t A309157 {a, b, c} // ColumnForm
%t A309157 a = Take[a, Length[c]]; b = Take[b, Length[c]];
%t A309157 Flatten[Transpose[{a, b, c}]] (* _Peter J. C. Moses_, Jul 04 2019 *)
%Y A309157 Cf. A326663 (3rd column),
%Y A309157 A101544 solves c(n) = a(n) + b(n),
%Y A309157 A326661 solves c(n) = a(n) + b(3n),
%Y A309157 A326662 solves c(n) = a(2n) + b(2n).
%K A309157 nonn,tabf,easy
%O A309157 1,2
%A A309157 _Clark Kimberling_, Jul 15 2019