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A294424 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) - 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

%I A294424 #5 Nov 01 2017 12:26:32
%S A294424 1,3,5,9,14,23,38,61,99,160,260,420,680,1100,1780,2880,4660,7540,
%T A294424 12201,19741,31942,51683,83625,135308,218933,354241,573174,927415,
%U A294424 1500589,2428004,3928593,6356597,10285191,16641788,26926979,43568767,70495746,114064513
%N A294424 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) - 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.
%C A294424 The complementary sequences a() and b() are uniquely determined by the titular equation and initial values.  See A294414 for a guide to related sequences.
%C A294424 Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.
%H A294424 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 A294424 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
%e A294424 a(2)  = a(1) + a(0) + b(1) - b(0) - 1 = 5
%e A294424 Complement: (b(n)) = (2, 4, 6, 7, 9, 11, 12, 13, 15,...)
%t A294424 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A294424 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
%t A294424 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] - b[n - 2] - 1;
%t A294424 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A294424 Table[a[n], {n, 0, 40}]  (* A294424 *)
%t A294424 Table[b[n], {n, 0, 10}]
%Y A294424 Cf. A293076, A293765, A294414.
%K A294424 nonn,easy
%O A294424 0,2
%A A294424 _Clark Kimberling_, Nov 01 2017