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A294553 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - n, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A294553 #7 Dec 31 2017 01:19:04
%S A294553 1,2,8,16,31,55,96,163,272,449,736,1201,1954,3174,5149,8345,13517,
%T A294553 21886,35428,57340,92795,150163,242987,393180,636198,1029410,1665641,
%U A294553 2695086,4360764,7055888,11416691,18472619,29889351,48362012,78251406,126613462
%N A294553 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - n, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A294553 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 A294553 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 A294553 a(0) = 1, a(1) = 2, b(0) = 3, so that
%e A294553 b(1) = 4 (least "new number")
%e A294553 a(2)  = a(1) + a(0) + b(1) + b(0) + b(1) - 2 = 8
%e A294553 Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 12, 13, 14, 15, 17, ...)
%t A294553 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A294553 a[0] = 1; a[1] = 2; b[0] = 3;
%t A294553 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2] - n;
%t A294553 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A294553 Table[a[n], {n, 0, 40}]  (* A294553 *)
%t A294553 Table[b[n], {n, 0, 10}]
%Y A294553 Cf. A001622, A294532.
%K A294553 nonn,easy
%O A294553 0,2
%A A294553 _Clark Kimberling_, Nov 15 2017