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A293076 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

%I A293076 #18 Sep 23 2020 04:11:23
%S A293076 1,3,6,13,24,44,76,129,215,355,582,951,1548,2515,4080,6613,10712,
%T A293076 17345,28078,45445,73546,119016,192588,311631,504247,815907,1320184,
%U A293076 2136122,3456338,5592493,9048865,14641393,23690294,38331724,62022056,100353819,162375915
%N A293076 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.
%C A293076 The complementary sequences a() and b() are uniquely determined by the titular equation and initial values, which for each sequence in the following guide are a(0) = 1, a(2) = 3, b(0) = 2, b(1) = 4:
%C A293076 A293076:  a(n) = a(n-1) + a(n-2) + b(n-2)
%C A293076 A293316:  a(n) = a(n-1) + a(n-2) + b(n-2) + 1
%C A293076 A293057:  a(n) = a(n-1) + a(n-2) + b(n-2) + 2
%C A293076 A293058:  a(n) = a(n-1) + a(n-2) + b(n-2) + 3
%C A293076 A293317:  a(n) = a(n-1) + a(n-2) + b(n-2) - 1
%C A293076 A293349:  a(n) = a(n-1) + a(n-2) + b(n-2) + n
%C A293076 A293350:  a(n) = a(n-1) + a(n-2) + b(n-2) + 2*n
%C A293076 A293351:  a(n) = a(n-1) + a(n-2) + b(n-2) + n - 1
%C A293076 A293357:  a(n) = a(n-1) + a(n-2) + b(n-2) + n + 1
%C A293076 Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.
%H A293076 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 A293076 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
%e A293076 a(2) = a(1) + a(0) + b(0) = 3 + 1 + 2 = 6;
%e A293076 a(3) = a(2) + a(1) + b(1) = 6 + 3 + 4 = 13.
%e A293076 Complement: (b(n)) = (2,4,5,7,8,9,10,11,12,14,...)
%t A293076 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A293076 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
%t A293076 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2];
%t A293076 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A293076 Table[a[n], {n, 0, 40}]  (* A293076 *)
%t A293076 Table[b[n], {n, 0, 10}]
%Y A293076 Cf. A001622 (golden ratio), A293358.
%K A293076 nonn,easy
%O A293076 0,2
%A A293076 _Clark Kimberling_, Oct 28 2017
%E A293076 Comments corrected by _Georg Fischer_, Sep 23 2020