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A294422 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 A294422 #5 Nov 01 2017 12:26:17
%S A294422 1,3,7,12,21,36,59,97,158,258,418,678,1098,1778,2878,4658,7538,12199,
%T A294422 19739,31940,51681,83623,135306,218931,354239,573172,927413,1500587,
%U A294422 2428002,3928591,6356595,10285189,16641786,26926977,43568765,70495744
%N A294422 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 A294422 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 A294422 Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.
%H A294422 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 A294422 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
%e A294422 a(2)  = a(1) + a(0) + b(1) - b(0) + 1 = 7
%e A294422 Complement: (b(n)) = (2, 4, 5, 6, 8, 9, 11, 13, ...)
%t A294422 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A294422 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
%t A294422 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] - b[n - 2] + 1;
%t A294422 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A294422 Table[a[n], {n, 0, 40}]  (* A294422 *)
%t A294422 Table[b[n], {n, 0, 10}]
%Y A294422 Cf. A293076, A293765, A294414.
%K A294422 nonn,easy
%O A294422 0,2
%A A294422 _Clark Kimberling_, Nov 01 2017