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A294384 Solution of the complementary equation a(n) = a(n-1)*b(n-2) - n, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4. 2

%I

%S 1,3,4,13,61,361,2521,20161,181441,1814401,19958401,239500801,

%T 3353011202,50295168017

%N Solution of the complementary equation a(n) = a(n-1)*b(n-2) - n, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

%C The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294381 for a guide to related sequences.

%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.pdf">Complementary equations</a>, J. Int. Seq. 19 (2007), 1-13.

%e a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that

%e a(2) = a(1)*b(0) - 2 = 4

%e Complement: (b(n)) = (2, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, ...)

%t mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;

%t a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;

%t a[n_] := a[n] = a[n - 1]*b[n - 2] - n;

%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, 40}] (* A294384 *)

%t Table[b[n], {n, 0, 10}]

%Y Cf. A293076, A293765, A294381.

%K nonn,more

%O 0,2

%A _Clark Kimberling_, Oct 29 2017

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Last modified July 14 10:35 EDT 2020. Contains 335722 sequences. (Running on oeis4.)