This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A295621 #7 Jun 08 2022 11:54:24
%S A295621 1,2,3,4,13,22,55,96,201,346,659,1117,2015,3372,5882,9752,16643,27411,
%T A295621 46093,75559,125754,205448,339432,553177,909097,1478897,2421000,
%U A295621 3933174,6420218,10419979,16972319,27525507,44762106,72554068,117844772,190931789,309833797
%N A295621 Solution of the complementary equation a(n) = a(n-1) + 3*a(n-2) -2*a(n-3) - 2*a(n-4) + b(n-3), where a(0) = 1, a(1) = 2, a(2) = 3, a(3) = 4, b(0) = 5, b(1) = 6, b(2) = 7, b(3) = 8, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A295621 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values.
%H A295621 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 A295621 a(0) = 1, a(1) = 2, a(2) = 3, a(3) = 4, b(0) = 5, b(1) = 6, b(2) = 7, b(3) = 8, so that
%e A295621 b(4) = 9 (least "new number")
%e A295621 a(4) = a(3) + 3*a(2) -2*a(1) - 2*a(0) + b(1) = 13
%e A295621 Complement: (b(n)) = (5, 6, 7, 8, 9, 10, 11, 12, 14, 15, ...)
%t A295621 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A295621 a[0] = 1; a[1] = 2; a[2] = 3; a[3] = 4;
%t A295621 b[0] = 5; b[1] = 6; b[2] = 7; b[3] = 8;
%t A295621 a[n_] := a[n] = a[n - 1] + 3*a[n - 2] - 2*a[n - 3] - 2 a[n - 4] + b[n - 3];
%t A295621 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A295621 z = 36; Table[a[n], {n, 0, z}] (* A295621 *)
%t A295621 Table[b[n], {n, 0, 20}] (*complement *)
%Y A295621 Cf. A001622, A000045, A295619, A295620.
%K A295621 nonn,easy
%O A295621 0,2
%A A295621 _Clark Kimberling_, Nov 25 2017
%E A295621 Typo in the definition corrected by _Georg Fischer_, Jun 08 2022