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A294537 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + n, where a(0) = 1, a(1) = 2, b(0) = 3.

%I A294537 #4 Nov 03 2017 09:54:13
%S A294537 1,2,8,17,34,62,109,187,314,521,857,1402,2285,3715,6030,9778,15843,
%T A294537 25658,41540,67239,108822,176106,284975,461130,746156,1207339,1953550,
%U A294537 3160946,5114555,8275562,13390180,21665808,35056056,56721934,91778062,148500070
%N A294537 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + n, where a(0) = 1, a(1) = 2, b(0) = 3.
%C A294537 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 A294537 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 A294537 a(0) = 1, a(1) = 2, b(0) = 3, so that
%e A294537 b(1) = 4 (least "new number")
%e A294537 a(2)  = a(1) + a(0) + b(0) + 2 = 8
%e A294537 Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, ...)
%t A294537 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A294537 a[0] = 1; a[1] = 3; b[0] = 2;
%t A294537 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2] + n;
%t A294537 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A294537 Table[a[n], {n, 0, 40}]  (* A294537 *)
%t A294537 Table[b[n], {n, 0, 10}]
%Y A294537 Cf. A001622, A294532.
%K A294537 nonn,easy
%O A294537 0,2
%A A294537 _Clark Kimberling_, Nov 03 2017