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A293765 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + 2, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

%I A293765 #7 Nov 02 2017 09:20:24
%S A293765 1,3,10,20,38,67,115,193,321,528,864,1408,2289,3715,6023,9758,15802,
%T A293765 25583,41409,67017,108452,175496,283976,459501,743507,1203039,1946578,
%U A293765 3149650,5096262,8245947,13342245,21588229,34930512,56518780,91449333,147968155
%N A293765 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + 2, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.
%C A293765 The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. The initial values of each sequence in the following guide are a(0) = 1, a(2) = 3, b(0) = 2, b(1) = 4:
%C A293765 Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.  See A293358 for a guide to related sequences.
%H A293765 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.pdf">Complementary equations</a>, J. Int. Seq. 19 (2007), 1-13.
%e A293765 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
%e A293765 a(2) = a(1) + a(0) + b(1) + 2 = 10;
%e A293765 Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, ...)
%t A293765 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A293765 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
%t A293765 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + 2;
%t A293765 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A293765 Table[a[n], {n, 0, 40}]  (* A293765 *)
%t A293765 Table[b[n], {n, 0, 10}]
%Y A293765 Cf. A001622 (golden ratio), A293765.
%K A293765 nonn,easy
%O A293765 0,2
%A A293765 _Clark Kimberling_, Oct 29 2017