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%I #4 Nov 15 2017 17:38:48
%S 1,2,5,10,17,29,48,80,130,212,344,558,904,1465,2371,3838,6211,10051,
%T 16264,26317,42583,68902,111487,180391,291881,472274,764157,1236433,
%U 2000592,3237027,5237621,8474650,13712273,22186925,35899200,58086127
%N Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) + 1, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
%C The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294532 for a guide to related sequences. Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).
%H 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 a(0) = 1, a(1) = 2, b(0) = 3, so that
%e b(1) = 4 (least "new number")
%e a(2) = a(1) + a(0) + b(1) - b(0) + 1 = 5
%e Complement: (b(n)) = (3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 18, ...)
%t mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t a[0] = 1; a[1] = 3; b[0] = 2;
%t a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] - b[n - 2] + 1;
%t b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t Table[a[n], {n, 0, 40}] (* A294562 *)
%t Table[b[n], {n, 0, 10}]
%Y Cf. A001622, A294532.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Nov 15 2017
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