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A295148 Solution of the complementary equation a(n) = a(n-1) + 2*a(n-2) + b(n-1), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A295148 #4 Nov 19 2017 19:06:25
%S A295148 1,3,9,20,44,91,187,379,764,1534,3075,6157,12322,24652,49313,98635,
%T A295148 197280,394571,789153,1578318,3156648,6313309,12626631,25253276,
%U A295148 50506566,101013147,202026309
%N A295148 Solution of the complementary equation a(n) = a(n-1) + 2*a(n-2) + b(n-1), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A295148 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.
%H A295148 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary equations</a>, J. Int. Seq. 19 (2007), 1-13.
%F A295148 a(n+1)/a(n) -> 2.
%e A295148 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4
%e A295148 a(2) = a(1) + 2*a(0) + b(1) = 9
%e A295148 Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, ...)
%t A295148 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A295148 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
%t A295148 a[n_] := a[n] = a[ n - 1] + 2 a[n - 2] + b[n - 1];
%t A295148 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A295148 Table[a[n], {n, 0, 18}]  (* A295148 *)
%t A295148 Table[b[n], {n, 0, 10}]
%Y A295148 Cf. A295053, A295145, A295146, A295147.
%K A295148 nonn,easy
%O A295148 0,2
%A A295148 _Clark Kimberling_, Nov 19 2017