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A294546 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + n, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A294546 #8 Nov 14 2017 22:19:28
%S A294546 1,2,9,19,38,69,121,207,347,575,945,1545,2517,4091,6639,10763,17438,
%T A294546 28239,45717,73998,119759,193803,313610,507463,821125,1328642,2149823,
%U A294546 3478523,5628406,9106991,14735461,23842518,38578047,62420635,100998755,163419465
%N A294546 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + n, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A294546 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 A294546 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 A294546 a(0) = 1, a(1) = 2, b(0) = 3, so that
%e A294546 b(1) = 4 (least "new number");
%e A294546 a(2) = a(1) + a(0) + b(1) + 2 = 9.
%e A294546 Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, ...).
%t A294546 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A294546 a[0] = 1; a[1] = 3; b[0] = 2;
%t A294546 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + n;
%t A294546 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A294546 Table[a[n], {n, 0, 40}]  (* A294546 *)
%t A294546 Table[b[n], {n, 0, 10}]
%Y A294546 Cf. A001622, A294532.
%K A294546 nonn,easy
%O A294546 0,2
%A A294546 _Clark Kimberling_, Nov 04 2017