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A294423 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) = 3, b(0) = 2, b(1) = 4.

%I A294423 #5 Nov 01 2017 12:26:25
%S A294423 1,3,8,15,28,49,85,142,236,388,635,1035,1684,2733,4432,7181,11630,
%T A294423 18829,30478,49327,79826,129175,209024,338223,547273,885522,1432822,
%U A294423 2318372,3751223,6069625,9820879,15890536,25711448,41602018,67313501,108915555,176229093
%N A294423 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) = 3, b(0) = 2, b(1) = 4.
%C A294423 The complementary sequences a() and b() are uniquely determined by the titular equation and initial values.  See A294414 for a guide to related sequences.
%C A294423 Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.
%H A294423 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 A294423 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
%e A294423 a(2) = a(1) + a(0) + b(1) - b(0)  + 2 = 8
%e A294423 Complement: (b(n)) = (2, 4, 5, 6, 7, 9, 11, 13, ...)
%t A294423 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A294423 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
%t A294423 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] - b[n - 2] + n;
%t A294423 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A294423 Table[a[n], {n, 0, 40}]  (* A294423 *)
%t A294423 Table[b[n], {n, 0, 10}]
%Y A294423 Cf. A293076, A293765, A294414.
%K A294423 nonn,easy
%O A294423 0,2
%A A294423 _Clark Kimberling_, Nov 01 2017