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A295060 Solution of the complementary equation a(n) = 2*a(n-1) - b(n-2), where a(0) = 3, a(1) = 5, b(0) = 1, b(1) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A295060 #4 Nov 18 2017 20:55:03
%S A295060 3,5,9,16,28,50,93,178,346,681,1350,2687,5360,10705,21393,42768,85517,
%T A295060 171014,342007,683992,1367961,2735898,5471771,10943516,21887005,
%U A295060 43773981,87547932,175095833,350191634
%N A295060 Solution of the complementary equation a(n) = 2*a(n-1) - b(n-2), where a(0) = 3, a(1) = 5, b(0) = 1, b(1) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A295060 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 A295060 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 A295060 a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3
%e A295060 a(2) = 2*a(1) - b(0) = 9
%e A295060 Complement: (b(n)) = (1, 2, 4, 6, 7, 8, 10, 11, 12, 13, 14, ...)
%t A295060 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A295060 a[0] = 3; a[1] = 5; b[0] = 1; b[1]=2;
%t A295060 a[n_] := a[n] = 2 a[n - 1] - b[n - 2];
%t A295060 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A295060 Table[a[n], {n, 0, 18}]  (* A295060 *)
%t A295060 Table[b[n], {n, 0, 10}]
%Y A295060 Cf. A295053.
%K A295060 nonn,easy
%O A295060 0,1
%A A295060 _Clark Kimberling_, Nov 18 2017