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A294554 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 2, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A294554 #8 Sep 27 2020 18:39:42
%S A294554 1,2,12,25,50,90,157,266,444,733,1203,1965,3199,5197,8431,13665,22135,
%T A294554 35841,58019,93905,151971,245925,397948,643928,1041933,1685920,
%U A294554 2727914,4413897,7141876,11555840,18697785,30253696,48951554,79205325,128156956,207362360
%N A294554 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 2, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A294554 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294532 for a guide to related sequences. Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).
%H A294554 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 A294554 a(0) = 1, a(1) = 2, b(0) = 3, so that
%e A294554 b(1) = 4 (least "new number")
%e A294554 a(2)  = a(1) + a(0) + b(1) + b(0) + 2 = 12
%e A294554 Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, ...)
%t A294554 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A294554 a[0] = 1; a[1] = 3; b[0] = 2;
%t A294554 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2] + 2;
%t A294554 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A294554 Table[a[n], {n, 0, 40}]  (* A294554 *)
%t A294554 Table[b[n], {n, 0, 10}]
%Y A294554 Cf. A001622, A294532.
%K A294554 nonn,easy
%O A294554 0,2
%A A294554 _Clark Kimberling_, Nov 15 2017
%E A294554 Definition corrected by _Georg Fischer_, Sep 27 2020