%I
%S 1,2,9,18,35,62,107,181,301,496,812,1324,2153,3495,5667,9183,14872,
%T 24078,38974,63077,102077,165181,267286,432496,699812,1132339,1832183,
%U 2964555,4796772,7761362,12558170,20319570,32877779,53197389,86075209,139272640
%N Solution of the complementary equation a(n) = a(n1) + a(n2) + b(n1) + 2, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
%C The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294532 for a guide to related sequences. Conjecture: a(n)/a(n1) > (1 + sqrt(5))/2 = golden ratio (A001622).
%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary equations</a>, J. Int. Seq. 19 (2007), 113.
%e a(0) = 1, a(1) = 2, b(0) = 3, so that
%e b(1) = 4 (least "new number");
%e a(2) = a(1) + a(0) + b(1) + 2 = 9.
%e Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, ...).
%t mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t a[0] = 1; a[1] = 3; b[0] = 2;
%t a[n_] := a[n] = a[n  1] + a[n  2] + b[n  1] + 2;
%t b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n  1}]]];
%t Table[a[n], {n, 0, 40}] (* A294543 *)
%t Table[b[n], {n, 0, 10}]
%Y Cf. A001622, A294532.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Nov 04 2017
