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A294414 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

%I A294414 #14 Sep 27 2020 18:37:33
%S A294414 1,3,10,22,43,78,136,231,387,641,1053,1721,2803,4555,7391,11981,19409,
%T A294414 31429,50879,82352,133278,215679,349008,564740,913803,1478600,2392462,
%U A294414 3871123,6263648,10134836,16398551,26533456,42932078,69465607,112397760,181863444
%N A294414 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.
%C A294414 The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. The initial values of each sequence in the following guide are a(0) = 1, a(2) = 3, b(0) = 2, b(1) = 4:
%C A294414 A294413:  a(n) = a(n-1) + a(n-2) - b(n-1) + 6
%C A294414 A294414:  a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2)
%C A294414 A294415:  a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 1
%C A294414 A294416:  a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n
%C A294414 A294417:  a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - n
%C A294414 A294418:  a(n) = a(n-1) + a(n-2) + b(n-1) + 2*b(n-2)
%C A294414 A294419:  a(n) = a(n-1) + a(n-2) + 2*b(n-1) + 2*b(n-2)
%C A294414 A294420:  a(n) = a(n-1) + a(n-2) + 2*b(n-1) + b(n-2)
%C A294414 A294421:  a(n) = a(n-1) + a(n-2) + 2*b(n-1) - b(n-2)
%C A294414 A294422:  a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) + 1
%C A294414 A294423:  a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) + n
%C A294414 A294424:  a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - 1
%C A294414 A294425:  a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - 2
%C A294414 A294426:  a(n) = a(n-1) + 2*a(n-2) + b(n-1) + b(n-2) - 3
%C A294414 Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.
%H A294414 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 A294414 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
%e A294414 a(2)  = a(1) + a(0) + b(1) + b(0) = 6
%e A294414 Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 11, 12, 13, ...)
%t A294414 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A294414 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
%t A294414 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2];
%t A294414 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A294414 Table[a[n], {n, 0, 40}]  (* A294414 *)
%t A294414 Table[b[n], {n, 0, 10}]
%Y A294414 Cf. A293076, A293765, A022940.
%K A294414 nonn,easy
%O A294414 0,2
%A A294414 _Clark Kimberling_, Oct 31 2017
%E A294414 Definition corrected by _Georg Fischer_, Sep 27 2020