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A293057 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + 2, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

%I A293057 #7 Nov 02 2017 09:19:17
%S A293057 1,3,8,17,32,57,98,166,276,455,745,1215,1976,3208,5202,8430,13653,
%T A293057 22105,35781,57910,93716,151652,245395,397075,642499,1039604,1682134,
%U A293057 2721770,4403937,7125742,11529715,18655494,30185247,48840780,79026067,127866888,206892997
%N A293057 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + 2, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.
%C A293057 The complementary sequences a() and b() are uniquely determined by the titular equation and initial values.  See A293076 for a guide to related sequences.
%C A293057 Conjecture:  a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.
%e A293057 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
%e A293057 a(2)  = a(1) + a(0) + b(0) + 2 = 8;
%e A293057 a(3) = a(2) + a(1) + b(1) + 1 = 17.
%e A293057 Complement: (b(n)) = (2,4,5,6,7,9,10,11,12,13,14,15,16,18,...)
%t A293057 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A293057 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
%t A293057 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2] + 2;
%t A293057 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A293057 Table[a[n], {n, 0, 40}]  (* A293316 *)
%t A293057 Table[b[n], {n, 0, 10}]
%Y A293057 Cf. A001622 (golden ratio), A293076.
%K A293057 nonn,easy
%O A293057 0,2
%A A293057 _Clark Kimberling_, Oct 28 2017