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A295065 Solution of the complementary equation a(n) = 8*a(n-3) + b(n-2), where a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A295065 #4 Nov 19 2017 10:43:02
%S A295065 1,3,5,12,30,47,104,249,386,843,2005,3102,6759,16056,24833,54090,
%T A295065 128467,198684,432741,1027758,1589495,3461952,8222089,12715986,
%U A295065 27695643,65776740,101727917
%N A295065 Solution of the complementary equation a(n) = 8*a(n-3) + b(n-2), where a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A295065 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 A295065 The sequence a(n+1)/a(n) appears to have three convergent subsequences, with limits 1.54..., 2.17..., 2.37...
%H A295065 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 A295065 a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4, b(2) = 6
%e A295065 a(3)  = 8*a(0) + b(1) = 12
%e A295065 Complement: (b(n)) = (2, 4, 6, 7, 8, 9, 10, 11, 13, 14, 15, ...)
%t A295065 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A295065 a[0] = 1; a[1] = 3; a[2] = 5; b[0] = 2;
%t A295065 a[n_] := a[n] = 8 a[n - 3] + b[n - 2];
%t A295065 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A295065 Table[a[n], {n, 0, 18}]  (* A295065 *)
%t A295065 Table[b[n], {n, 0, 10}]
%Y A295065 Cf. A295053, A295064.
%K A295065 nonn,easy
%O A295065 0,2
%A A295065 _Clark Kimberling_, Nov 19 2017