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A293317 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) - 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

%I A293317 #7 Nov 02 2017 09:19:31
%S A293317 1,3,5,11,21,38,66,112,187,310,509,832,1355,2202,3573,5792,9383,15194,
%T A293317 24598,39814,64435,104273,168733,273032,441792,714852,1156673,1871555,
%U A293317 3028259,4899846,7928138,12828018,20756191,33584245,54340474,87924758,142265272
%N A293317 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) - 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.
%C A293317 The complementary sequences a() and b() are uniquely determined by the titular equation and initial values.  See A293076 for a guide to related sequences.
%C A293317 Conjecture:  a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.
%e A293317 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
%e A293317 a(2)  = a(1) + a(0) + b(0) -1 = 5;
%e A293317 a(3) = a(2) + a(1) + b(1) - 1 = 11.
%e A293317 Complement: (b(n)) = (2, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15,...)
%t A293317 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A293317 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
%t A293317 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2] - 1;
%t A293317 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A293317 Table[a[n], {n, 0, 40}]  (* A293317 *)
%t A293317 Table[b[n], {n, 0, 10}]
%Y A293317 Cf. A001622 (golden ratio), A293076.
%K A293317 nonn,easy
%O A293317 0,2
%A A293317 _Clark Kimberling_, Oct 28 2017