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A294866 Solution of the complementary equation a(n) = 2*a(n-1) - a(n-2) + b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A294866 #7 Sep 27 2020 18:37:16
%S A294866 1,2,7,17,33,57,90,133,187,253,332,425,533,657,799,960,1141,1343,1567,
%T A294866 1814,2085,2381,2703,3052,3429,3835,4271,4738,5237,5770,6338,6942,
%U A294866 7583,8262,8980,9738,10537,11378,12262,13190,14163,15182,16248,17362,18525,19738
%N A294866 Solution of the complementary equation a(n) = 2*a(n-1) - a(n-2) + b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A294866 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 A294866 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 A294866 a(0) = 1, a(1) = 2, b(0) = 3
%e A294866 b(1) = 4 (least "new number")
%e A294866 a(2) = 2*a(1) - a(0) + b(1) = 7
%e A294866 Complement: (b(n)) = (3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, ...)
%t A294866 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A294866 a[0] = 1; a[1] = 2; b[0] = 3;
%t A294866 a[n_] := a[n] = 2*a[n - 1] - a[n - 2] + b[n - 1];
%t A294866 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A294866 Table[a[n], {n, 0, 18}]  (* A294866 *)
%t A294866 Table[b[n], {n, 0, 10}]
%Y A294866 Cf. A294860.
%K A294866 nonn,easy
%O A294866 0,2
%A A294866 _Clark Kimberling_, Nov 16 2017
%E A294866 Definition corrected by _Georg Fischer_, Sep 27 2020