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A295058 Solution of the complementary equation a(n) = 2*a(n-1) - b(n-1), where a(0) = 3, a(1) = 5, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A295058 #4 Nov 18 2017 20:54:50
%S A295058 3,5,8,12,18,29,49,88,165,317,620,1225,2434,4851,9683,19346,38671,
%T A295058 77320,154617,309210,618395,1236764,2473501,4946974,9893918,19787805,
%U A295058 39575578,79151123,158302212,316604389,633208742
%N A295058 Solution of the complementary equation a(n) = 2*a(n-1) - b(n-1), where a(0) = 3, a(1) = 5, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A295058 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 A295058 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 A295058 a(0) = 3, a(1) = 5, b(0) = 1
%e A295058 b(1) = 2 (least "new number")
%e A295058 a(2) = 2*a(1) - b(1) = 8
%e A295058 Complement: (b(n)) = (1, 2, 4, 6, 7, 9, 10, 11, 13, 14, 15, ...)
%t A295058 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A295058 a[0] = 3; a[1] = 5; b[0] = 1;
%t A295058 a[n_] := a[n] = 2 a[n - 1] - b[n - 1];
%t A295058 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A295058 Table[a[n], {n, 0, 18}]  (* A295058 *)
%t A295058 Table[b[n], {n, 0, 10}]
%Y A295058 Cf. A295053.
%K A295058 nonn,easy
%O A295058 0,1
%A A295058 _Clark Kimberling_, Nov 18 2017