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A295059 Solution of the complementary equation a(n) = 2*a(n-1) + b(n-2), where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A295059 #4 Nov 18 2017 20:54:56
%S A295059 1,4,10,23,51,108,223,454,917,1845,3702,7417,14848,29711,59438,118893,
%T A295059 237804,475627,951274,1902569,3805160,7610344,15220713,30441452,
%U A295059 60882931,121765890,243531809,487063648
%N A295059 Solution of the complementary equation a(n) = 2*a(n-1) + b(n-2), where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A295059 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 A295059 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 A295059 a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3
%e A295059 a(2) = 2*a(1) + b(0) = 10
%e A295059 Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, ...)
%t A295059 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A295059 a[0] = 2; a[1] = 5; b[0] = 1;
%t A295059 a[n_] := a[n] = 2 a[n - 1] + b[n - 1];
%t A295059 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A295059 Table[a[n], {n, 0, 18}]  (* A295059 *)
%t A295059 Table[b[n], {n, 0, 10}]
%Y A295059 Cf. A295053.
%K A295059 nonn,easy
%O A295059 0,2
%A A295059 _Clark Kimberling_, Nov 18 2017