A294383 - OEIS

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A294383

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

1, 3, 7, 29, 146, 877, 7017, 63154, 631541, 6946952, 83363425, 1083724526, 15172143365

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OFFSET

0,2

COMMENTS

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

LINKS

Table of n, a(n) for n=0..12.

Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

EXAMPLE

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

a(2) = a(1)*b(0) + 1 = 7

Complement: (b(n)) = (2, 4, 5, 6, 8, 9, 10, 12, 13, 14, 15, 16, ...)

MATHEMATICA

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

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

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

b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];