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A295054 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(1) + b(2) + ... + b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A295054 #4 Nov 18 2017 20:54:14
%S A295054 1,2,7,18,40,81,153,276,482,823,1383,2298,3788,6209,10137,16505,26821,
%T A295054 43526,70569,114340,185178,299812,485310,785469,1271154,2057027,
%U A295054 3328615,5386107,8715219,14101856,22817639,36920094,59738368,96659134
%N A295054 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(1) + b(2) + ... + b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A295054 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 A295054 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 A295054 a(0) = 1, a(1) = 2, b(0) = 3
%e A295054 b(1) = 4 (least "new number")
%e A295054 a(2) = a(1) + a(0) + b(1) = Complement: (b(n)) = (3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, ...)
%t A295054 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A295054 a[0] = 1; a[1] = 2; b[0] = 3;
%t A295054 a[n_] := a[n] = a[n - 1] + a[n - 2] + Sum[b[k], {k, 1, n - 1}];
%t A295054 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A295054 Table[a[n], {n, 0, 18}]  (* A295054 *)
%t A295054 Table[b[n], {n, 0, 10}]
%Y A295054 Cf. A295053.
%K A295054 nonn,easy
%O A295054 0,2
%A A295054 _Clark Kimberling_, Nov 18 2017