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A295143 Solution of the complementary equation a(n) = 2*a(n-1) + a(n-2) + b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A295143 #4 Nov 19 2017 19:05:44
%S A295143 1,2,9,25,65,162,397,966,2340,5658,13669,33010,79704,192434,464589,
%T A295143 1121630,2707868,6537386,15782661,38102730,91988144,222079042
%N A295143 Solution of the complementary equation a(n) = 2*a(n-1) + a(n-2) + b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A295143 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 A295143 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 A295143 a(n+1)/a(n) -> 1 + sqrt(2).
%e A295143 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4
%e A295143 a(2) =2*a(1) + a(0) + b(1) = 9
%e A295143 Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, ...)
%t A295143 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A295143 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
%t A295143 a[n_] := a[n] = 2 a[ n - 1] + a[n - 2] + b[n - 1];
%t A295143 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A295143 Table[a[n], {n, 0, 18}]  (* A295143 *)
%t A295143 Table[b[n], {n, 0, 10}]
%Y A295143 Cf. A295053, A295141, A295142, A295144.
%K A295143 nonn,easy
%O A295143 0,2
%A A295143 _Clark Kimberling_, Nov 19 2017