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A294872 Solution of the complementary equation a(n) = 2*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 A294872 #4 Nov 18 2017 20:53:58
%S A294872 1,2,9,24,49,86,137,205,292,400,531,687,870,1082,1325,1601,1912,2260,
%T A294872 2647,3075,3546,4063,4628,5243,5910,6631,7408,8243,9138,10095,11116,
%U A294872 12203,13358,14583,15880,17251,18698,20223,21828,23515,25286,27143,29088,31123
%N A294872 Solution of the complementary equation a(n) = 2*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 A294872 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 A294872 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 A294872 a(0) = 1, a(1) = 2, b(0) = 3
%e A294872 b(1) = 4 (least "new number")
%e A294872 a(2) = 2*a(1) - a(0) + b(1) + 2 = 9
%e A294872 Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, ...)
%t A294872 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A294872 a[0] = 1; a[1] = 2; b[0] = 3;
%t A294872 a[n_] := a[n] = 2 a[n - 1] - a[n - 2] + b[n - 1] + n;
%t A294872 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A294872 Table[a[n], {n, 0, 18}]  (* A294872 *)
%t A294872 Table[b[n], {n, 0, 10}]
%Y A294872 Cf. A294860.
%K A294872 nonn,easy
%O A294872 0,2
%A A294872 _Clark Kimberling_, Nov 18 2017