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A294870 Solution of the complementary equation a(n) = 2*a(n-1) - a(n-2) + b(n-1) + 2, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A294870 #4 Nov 18 2017 09:05:25
%S A294870 1,2,9,23,45,76,117,170,236,316,411,522,650,796,961,1146,1352,1580,
%T A294870 1831,2106,2407,2735,3091,3476,3891,4337,4815,5326,5871,6451,7067,
%U A294870 7720,8411,9141,9911,10722,11575,12471,13411,14396,15427,16506,17634,18812,20041
%N A294870 Solution of the complementary equation a(n) = 2*a(n-1) - a(n-2) + b(n-1) + 2, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A294870 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 A294870 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 A294870 a(0) = 1, a(1) = 2, b(0) = 3
%e A294870 b(1) = 4 (least "new number")
%e A294870 a(2) = 2*a(1) - a(0) + b(1) + 2 = 9
%e A294870 Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, ...)
%e A294870 m
%t A294870 ex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A294870 a[0] = 1; a[1] = 2; b[0] = 3;
%t A294870 a[n_] := a[n] = 2 a[n - 1] - a[n - 2] + b[n - 1] + 2;
%t A294870 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A294870 Table[a[n], {n, 0, 18}]  (* A294870 *)
%t A294870 Table[b[n], {n, 0, 10}]
%Y A294870 Cf. A294860.
%K A294870 nonn,easy
%O A294870 0,2
%A A294870 _Clark Kimberling_, Nov 16 2017