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A295056 Solution of the complementary equation a(n) = 2*a(n-1) + b(n-1), where a(0) = 1, a(1) = 4, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A295056 #4 Nov 18 2017 20:54:28
%S A295056 1,4,11,27,60,127,262,533,1076,2164,4341,8696,17407,34830,69677,
%T A295056 139372,278763,557546,1115113,2230248,4460519,8921062,17842149,
%U A295056 35684324,71368676,142737381,285474792,570949615
%N A295056 Solution of the complementary equation a(n) = 2*a(n-1) + b(n-1), where a(0) = 1, a(1) = 4, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A295056 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 A295056 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 A295056 a(0) = 1, a(1) = 4, b(0) = 2
%e A295056 b(1) = 3 (least "new number")
%e A295056 a(2) = 2*a(1) + b(1) = 11
%e A295056 Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, ...)
%t A295056 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A295056 a[0] = 1; a[1] = 4; b[0] = 2;
%t A295056 a[n_] := a[n] = 2 a[n - 1] + b[n - 1];
%t A295056 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A295056 Table[a[n], {n, 0, 18}]  (* A295056 *)
%t A295056 Table[b[n], {n, 0, 10}]
%Y A295056 Cf. A295053.
%K A295056 nonn,easy
%O A295056 0,2
%A A295056 _Clark Kimberling_, Nov 18 2017