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A296217 Solution of the complementary equation a(n) = a(1)*b(n-2) + a(2)*b(n-3) + ... + a(n-1)*b(0), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A296217 #4 Dec 09 2017 19:28:38
%S A296217 1,2,6,26,112,484,2088,9008,38862,167658,723308,3120486,13462360,
%T A296217 58079138,250564260,1080981064,4663554414,20119445656,86799050160,
%U A296217 374467330636,1615522076050,6969664279584,30068434774274,129720849313094,559639996988064,2414391579204576
%N A296217 Solution of the complementary equation a(n) = a(1)*b(n-2) + a(2)*b(n-3) + ... + a(n-1)*b(0), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A296217 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295862 for a guide to related sequences.
%H A296217 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 A296217 a(0) =1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 4
%e A296217 a(2) = a(1)*b(0) = 2
%e A296217 Complement: (b(n)) = (3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...)
%t A296217 a[0] = 1; a[1] = 2; b[0] = 3;
%t A296217 a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 1, n - 1}];
%t A296217 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A296217 u = Table[a[n], {n, 0, 200}];  (* A296217 *)
%t A296217 Table[b[n], {n, 0, 20}]
%t A296217 N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
%t A296217 RealDigits[Last[t], 10][[1]] (* A296218 *)
%Y A296217 Cf. A296000, A296218.
%K A296217 nonn,easy
%O A296217 0,2
%A A296217 _Clark Kimberling_, Dec 08 2017