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%I #7 Aug 27 2021 21:14:33
%S 1,2,3,11,27,59,116,215,383,663,1125,1882,3117,5126,8388,13678,22250,
%T 36133,58610,94993,153877,249169,403371,652893,1056646,1709951,
%U 2767040,4477466,7245014,11723022,18968613,30692248,49661511,80354447,130016685,210371899
%N Solution of the complementary equation a(n) = 2*a(n-1) - a(n-3) + b(n-1), where a(0) = 1, a(1) = 2, a(2) = 3, b(0) = 4, b(1) = 5, b(2) = 6, and (a(n)) and (b(n)) are increasing complementary sequences.
%C The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. Guide to related sequences:
%C A295613: a(n) = 2*a(n-1) - a(n-3) + b(n-1); a(0) = 1, a(1) = 2, a(2) = 3, b(0) = 4, b(1) = 5, b(2) = 6.
%C A295614: a(n) = 2*a(n-1) - a(n-3) + b(n-1); a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4, b(2) = 6.
%C A295615: a(n) = 2*a(n-1) - a(n-3) + b(n-1); a(0) = 2, a(1) = 4, a(2) = 6, b(0) = 1, b(1) = 3, b(2) = 5.
%C A295616: a(n) = 2*a(n-1) - a(n-3) + b(n-2); a(0) = 1, a(1) = 2, a(2) = 3, b(0) = 4, b(1) = 5, b(2) = 6.
%C A295617: a(n) = 2*a(n-1) - a(n-3) + b(n-2); a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4, b(2) = 6.
%C A295618: a(n) = 2*a(n-1) - a(n-3) + b(n-2); a(0) = 2, a(1) = 4, a(2) = 6, b(0) = 1, b(1) = 3, b(2) = 5.
%C a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622), so that a( ) has the growth rate of the Fibonacci numbers (A000045).
%H 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 a(0) = 1, a(1) = 2, a(2) = 3, b(0) = 4, b(1) = 5, b(2) = 6, so that
%e b(3) = 7 (least "new number")
%e a(3) = 2*a(2) - a(0) + b(2) = 11
%e Complement: (b(n)) = (4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, ...)
%t mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t a[0] = 1; a[1] = 2; a[2] = 3; b[0] = 4; b[1] = 5; b[2] = 6;
%t a[n_] := a[n] = 2 a[n - 1] - a[n - 3] + b[n - 1];
%t b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t Table[a[n], {n, 0, 30}] (* A295613 *)
%t Table[b[n], {n, 0, 20}] (* complement *)
%Y Cf. A001622, A000045.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Nov 25 2017
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