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A299547 Solution a( ) of the complementary equation a(n) = b(n-1) + b(n-2) + ... + b(0), where a(0) = 1, a(1) = 2, a(2) = 3; see Comments.

%I A299547 #12 Jan 05 2025 19:51:41
%S A299547 1,2,3,12,18,25,33,42,52,63,76,90,105,121,138,157,177,198,220,243,267,
%T A299547 293,320,348,377,407,438,470,504,539,575,612,650,689,729,770,813,857,
%U A299547 902,948,995,1043,1092,1142,1193,1246,1300,1355,1411,1468,1526,1585
%N A299547 Solution a( ) of the complementary equation a(n) = b(n-1) + b(n-2) + ... + b(0), where a(0) = 1, a(1) = 2, a(2) = 3; see Comments.
%C A299547 From the Bode-Harborth-Kimberling link:
%C A299547 a(n) = b(n-1) + b(n-2) + ... + b(0) for n > 3;
%C A299547 b(0) = least positive integer not in {a(0),a(1),a(2)};
%C A299547 b(n) = least positive integer not in {a(0),...,a(n),b(0),...,b(n-1)} for n > 1.
%C A299547 Note that (b(n)) is strictly increasing and is the complement of (a(n)).
%C A299547 See A022424 for a guide to related sequences.
%H A299547 J-P. Bode, H. Harborth, C. Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/45-3/bode.pdf">Complementary Fibonacci sequences</a>, Fibonacci Quarterly 45 (2007), 254-264.
%t A299547 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A299547 a[0] = 1; a[1] = 2; a[2] = 3; b[0] = 4;
%t A299547 a[n_] := a[n] = Sum[b[k], {k, 0, n - 1}];
%t A299547 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A299547 Table[a[n], {n, 0, 100}]    (* A299547 *)
%Y A299547 Cf. A022424.
%K A299547 nonn,easy
%O A299547 0,2
%A A299547 _Clark Kimberling_, Mar 01 2018
%E A299547 Definition corrected by _Georg Fischer_, Sep 28 2020