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A295067 Solution of the complementary equation a(n) = 2*a(n-2) + b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A295067 #4 Nov 19 2017 10:43:19
%S A295067 1,3,4,11,14,29,36,67,82,146,177,307,370,631,758,1281,1536,2583,3094,
%T A295067 5189,6212,10403,12450,20833,24928,41696,49887,83424,99807,166882,
%U A295067 199649,333801,399336,667641,798712,1335323,1597466,2670689,3194976,5341423,6389998
%N A295067 Solution of the complementary equation a(n) = 2*a(n-2) + b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A295067 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295053 for a guide to related sequences.
%C A295067 The sequence a(n+1)/a(n) appears to have two convergent subsequences, with limits 1.19..., 1.67...
%H A295067 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 A295067 a(0) = 1, a(1) = 3, a(2) = 3, b(0) = 2, b(1) = 5
%e A295067 a(2) = 2*a(0) + b(0) = 4
%e A295067 Complement: (b(n)) = (2, 5, 6, 7, 8, 9, 10, 12, 13, 15, ... )
%t A295067 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A295067 a[0] = 1; a[1] = 3; b[0] = 2; b[1]=5;
%t A295067 a[n_] := a[n] = 2 a[n - 2] + b[n - 2];
%t A295067 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A295067 Table[a[n], {n, 0, 18}]  (* A295067 *)
%t A295067 Table[b[n], {n, 0, 10}]
%Y A295067 Cf. A295053, A295066.
%K A295067 nonn,easy
%O A295067 0,2
%A A295067 _Clark Kimberling_, Nov 19 2017