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A293357 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + n +1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

%I A293357 #8 Nov 02 2017 09:19:59
%S A293357 1,3,9,20,39,71,124,211,354,586,963,1574,2564,4167,6762,10962,17759,
%T A293357 28758,46557,75357,121958,197361,319367,516778,836197,1353029,2189282,
%U A293357 3542369,5731711,9274142,15005917,24280125,39286110,63566305,102852487,166418866
%N A293357 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + n +1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.
%C A293357 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 A293357 Conjecture:  a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.
%e A293357 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
%e A293357 a(2)  = a(1) + a(0) + b(0) + 3 = 9;
%e A293357 a(3) = a(2) + a(1) + b(1) + 4 = 20.
%e A293357 Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14,...)
%t A293357 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A293357 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
%t A293357 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2] + n + 1;
%t A293357 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A293357 Table[a[n], {n, 0, 40}]  (* A293357 *)
%t A293357 Table[b[n], {n, 0, 10}]
%Y A293357 Cf. A001622 (golden ratio), A293076.
%K A293357 nonn,easy
%O A293357 0,2
%A A293357 _Clark Kimberling_, Oct 28 2017