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A295057 Solution of the complementary equation a(n) = 2*a(n-1) + b(n-1), where a(0) = 2, a(1) = 5, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A295057 #4 Nov 18 2017 20:54:41
%S A295057 2,5,13,30,66,139,286,581,1172,2355,4722,9458,18931,37878,75773,
%T A295057 151564,303147,606314,1212649,2425320,4850663,9701350,19402725,
%U A295057 38805476,77610979,155221986,310444001,620888033
%N A295057 Solution of the complementary equation a(n) = 2*a(n-1) + b(n-1), where a(0) = 2, a(1) = 5, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A295057 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 A295057 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 A295057 a(0) = 2, a(1) = 5, b(0) = 1
%e A295057 b(1) = 3 (least "new number")
%e A295057 a(2) = 2*a(1) + b(1) = 13
%e A295057 Complement: (b(n)) = (1, 3, 4, 6, 7, 8, 9, 10, 11, 12, 14, ...)
%t A295057 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A295057 a[0] = 2; a[1] = 5; b[0] = 1;
%t A295057 a[n_] := a[n] = 2 a[n - 1] + b[n - 1];
%t A295057 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A295057 Table[a[n], {n, 0, 18}]  (* A295057 *)
%t A295057 Table[b[n], {n, 0, 10}]
%Y A295057 Cf. A295053.
%K A295057 nonn,easy
%O A295057 0,1
%A A295057 _Clark Kimberling_, Nov 18 2017