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A294418 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + 2*b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

%I A294418 #9 Nov 01 2017 20:54:19
%S A294418 1,3,12,28,56,103,181,309,518,858,1411,2309,3763,6118,9930,16100,
%T A294418 26085,42243,68389,110696,179152,289918,469143,759137,1228359,1987579,
%U A294418 3216026,5203696,8419816,13623609,22043525,35667237,57710868,93378214,151089194,244467523
%N A294418 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + 2*b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.
%C A294418 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 A294418 Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.
%H A294418 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 A294418 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
%e A294418 a(2)  = a(1) + a(0) + b(1) + 2*b(0) = 12
%e A294418 Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14,...)
%t A294418 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A294418 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
%t A294418 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + 2 b[n - 2];
%t A294418 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A294418 Table[a[n], {n, 0, 40}]  (* A294418 *)
%t A294418 Table[b[n], {n, 0, 10}]
%Y A294418 Cf. A293076, A293765, A294414.
%K A294418 nonn,easy
%O A294418 0,2
%A A294418 _Clark Kimberling_, Oct 31 2017