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A295145 Solution of the complementary equation a(n) = a(n-1) + 2*a(n-2) + b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A295145 #4 Nov 19 2017 19:05:59
%S A295145 1,2,7,15,34,70,146,295,597,1198,2404,4813,9635,19277,38564,77136,
%T A295145 154283,308575,617162,1234334,2468681,4937373,9874760,19749532,
%U A295145 39499079,78998171,157996358
%N A295145 Solution of the complementary equation a(n) = a(n-1) + 2*a(n-2) + b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A295145 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 A295145 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 A295145 a(n+1)/a(n) -> 2.
%e A295145 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4
%e A295145 a(2) = a(1) + 2*a(0) + b(0) = 7
%e A295145 Complement: (b(n)) = (3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, ...)
%t A295145 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A295145 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
%t A295145 a[n_] := a[n] = a[ n - 1] + 2 a[n - 2] + b[n - 2];
%t A295145 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A295145 Table[a[n], {n, 0, 18}]  (* A295145 *)
%t A295145 Table[b[n], {n, 0, 10}]
%Y A295145 Cf. A295053, A295146, A295147, A295148.
%K A295145 nonn,easy
%O A295145 0,2
%A A295145 _Clark Kimberling_, Nov 19 2017