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A295061 Solution of the complementary equation a(n) = 4*a(n-2) + b(n-1), where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A295061 #4 Nov 18 2017 20:55:10
%S A295061 1,3,8,17,38,75,161,310,655,1252,2633,5022,10547,20104,42206,80435,
%T A295061 168844,321761,675398,1287067,2701616,5148293,10806490,20593199,
%U A295061 43225988,82372825,172903982
%N A295061 Solution of the complementary equation a(n) = 4*a(n-2) + b(n-1), where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A295061 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.
%C A295061 The sequence a(n+1)/a(n) appears to have two convergent subsequences, with limits 1.09... and 2.09... .
%H A295061 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 A295061 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4
%e A295061 a(2) = 4*a(0) + b(1) = 8
%e A295061 Complement: (b(n)) = (2, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, ...)
%t A295061 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A295061 a[0] = 1; a[1] = 3; b[0] = 2;
%t A295061 a[n_] := a[n] = 4 a[n - 2] + b[n - 1];
%t A295061 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A295061 Table[a[n], {n, 0, 18}]  (* A295061 *)
%t A295061 Table[b[n], {n, 0, 10}]
%Y A295061 Cf. A295053.
%K A295061 nonn,easy
%O A295061 0,2
%A A295061 _Clark Kimberling_, Nov 18 2017