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A294420 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + 2*b(n-1) + b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

%I A294420 #9 Nov 01 2017 20:54:32
%S A294420 1,3,14,31,62,113,198,337,564,933,1532,2503,4078,6628,10756,17437,
%T A294420 28249,45745,74056,119866,193990,313927,507991,821995,1330066,2152144,
%U A294420 3482296,5634529,9116919,14751546,23868566,38620216,62488889,101109215,163598217,264707548
%N A294420 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + 2*b(n-1) + b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.
%C A294420 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 A294420 Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.
%H A294420 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 A294420 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
%e A294420 a(2)  = a(1) + a(0) + 2*b(1) + b(n-2) = 14
%e A294420 Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17,...)
%t A294420 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A294420 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
%t A294420 a[n_] := a[n] = a[n - 1] + a[n - 2] + 2 b[n - 1] + b[n - 2];
%t A294420 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A294420 Table[a[n], {n, 0, 40}]  (* A294420 *)
%t A294420 Table[b[n], {n, 0, 10}]
%Y A294420 Cf. A293076, A293765, A294414.
%K A294420 nonn,easy
%O A294420 0,2
%A A294420 _Clark Kimberling_, Oct 31 2017