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A294541 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A294541 #10 Nov 14 2017 22:16:56
%S A294541 1,2,7,14,27,49,85,144,240,396,649,1060,1725,2802,4545,7366,11931,
%T A294541 19318,31271,50612,81907,132544,214477,347049,561555,908634,1470220,
%U A294541 2378886,3849139,6228059,10077233,16305328,26382598,42687964,69070601,111758605
%N A294541 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A294541 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(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).
%H A294541 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary equations</a>, J. Int. Seq. 19 (2007), 1-13.
%e A294541 a(0) = 1, a(1) = 2, b(0) = 3, so that
%e A294541 b(1) = 4 (least "new number");
%e A294541 a(2) = a(1) + a(0) + b(1) = 7;
%e A294541 Complement: (b(n)) = (3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, ...).
%t A294541 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A294541 a[0] = 1; a[1] = 3; b[0] = 2;
%t A294541 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1];
%t A294541 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A294541 Table[a[n], {n, 0, 40}]  (* A294541 *)
%t A294541 Table[b[n], {n, 0, 10}]
%Y A294541 Cf. A001622, A294532.
%K A294541 nonn,easy
%O A294541 0,2
%A A294541 _Clark Kimberling_, Nov 04 2017