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A294867 Solution of the complementary equation a(n) = 2*a(n-1) - a(n-2) + b(n-1) -1, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A294867 #4 Nov 18 2017 09:06:28
%S A294867 1,2,6,14,28,49,78,116,164,223,294,379,479,595,728,879,1049,1239,1450,
%T A294867 1683,1939,2219,2524,2855,3214,3602,4020,4469,4950,5464,6012,6595,
%U A294867 7214,7870,8564,9297,10070,10884,11740,12639,13582,14570,15604,16685,17815,18995
%N A294867 Solution of the complementary equation a(n) = 2*a(n-1) - a(n-2) + b(n-1) -1, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A294867 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294860 for a guide to related sequences.
%H A294867 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 A294867 a(0) = 1, a(1) = 2, b(0) = 3
%e A294867 b(1) = 4 (least "new number")
%e A294867 a(2) = 2*a(1) - a(0) + b(1) - 1 = 6
%e A294867 Complement: (b(n)) = (3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 15, ...)
%t A294867 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A294867 a[0] = 1; a[1] = 2; b[0] = 3;
%t A294867 a[n_] := a[n] = 2 a[n - 1] - a[n - 2] + b[n - 1] - 1;
%t A294867 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A294867 Table[a[n], {n, 0, 18}]  (* A294867 *)
%t A294867 Table[b[n], {n, 0, 10}]
%Y A294867 Cf. A294860.
%K A294867 nonn,easy
%O A294867 0,2
%A A294867 _Clark Kimberling_, Nov 16 2017