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A294869 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 A294869 #4 Nov 18 2017 09:06:43
%S A294869 1,2,8,20,39,66,103,151,211,284,371,473,591,726,879,1051,1243,1457,
%T A294869 1694,1955,2241,2553,2892,3259,3655,4081,4538,5027,5549,6105,6696,
%U A294869 7323,7987,8689,9430,10212,11036,11903,12814,13770,14772,15821,16918,18064,19260
%N A294869 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 A294869 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 A294869 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 A294869 a(0) = 1, a(1) = 2, b(0) = 3
%e A294869 b(1) = 4 (least "new number")
%e A294869 a(2) = 2*a(1) - a(0) + b(1) + 1 = 8
%e A294869 Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, ...)
%t A294869 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A294869 a[0] = 1; a[1] = 2; b[0] = 3;
%t A294869 a[n_] := a[n] = 2 a[n - 1] - a[n - 2] + b[n - 1] + 1;
%t A294869 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A294869 Table[a[n], {n, 0, 18}]  (* A294869 *)
%t A294869 Table[b[n], {n, 0, 10}]
%Y A294869 Cf. A294860.
%K A294869 nonn,easy
%O A294869 0,2
%A A294869 _Clark Kimberling_, Nov 16 2017