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A295998 Solution of the complementary equation a(n) = 2*a(n-2) + b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A295998 #5 Dec 02 2017 21:00:51
%S A295998 1,2,5,8,16,23,41,56,93,124,199,262,413,541,844,1101,1708,2223,3438,
%T A295998 4470,6901,8966,13829,17960,27687,35950,55405,71932,110843,143898,
%U A295998 221721,287832,443479,575702,886997,1151444,1774036,2302931,3548116,4605907,7096278
%N A295998 Solution of the complementary equation a(n) = 2*a(n-2) + b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A295998 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values.  a(n)/a(n-1) -> 1.298123759410105...
%C A295998 See A295860 for a guide to related sequences.
%H A295998 Clark Kimberling, <a href="/A295998/b295998.txt">Table of n, a(n) for n = 0..999</a>
%H A295998 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary equations</a>, J. Int. Seq. 19 (2007), 1-13.
%F A295998 a(0) = 1, a(1) = 2, b(0) = 3, so that a(2) = 5, b(1) = 4.
%F A295998 Complement: (b(n)) = (3, 4, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, ...)
%t A295998 mex[t_] := NestWhile[# + 1 &, 1, MemberQ[t, #] &];
%t A295998 a[0] = 1; a[1] = 2; b[0] = 3;
%t A295998 a[n_] := a[n] = 2 a[n - 2] + b[n - 2];  (* A295998 *)
%t A295998 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A295998 Table[a[n], {n, 0, 100}];
%t A295998 Table[b[n], {n, 0, 30}]
%Y A295998 Cf. A001622, A000045, A294860.
%K A295998 nonn,easy
%O A295998 0,2
%A A295998 _Clark Kimberling_, Dec 02 2017