A294381 Solution of the complementary equation a(n) = a(n-1)*b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 6, 24, 120, 840, 6720, 60480, 604800, 6652800, 79833600, 1037836800, 14529715200, 217945728000, 3487131648000, 59281238016000, 1067062284288000, 20274183401472000, 405483668029440000, 8515157028618240000, 187333454629601280000, 4308669456480829440000, 107716736412020736000000

Offset

0,2

Comments

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. The initial values of each sequence in the following guide are a(0) = 1, a(2) = 3, b(0) = 2, b(1) = 4:

A294381: a(n) = a(n-1)*b(n-2)

A294382: a(n) = a(n-1)*b(n-2) - 1

A294383: a(n) = a(n-1)*b(n-2) + 1

A294384: a(n) = a(n-1)*b(n-2) - n

A294385: a(n) = a(n-1)*b(n-2) + n

Links

Jack W Grahl, Table of n, a(n) for n = 0..49

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) = 6.
Complement: (b(n)) = (2, 4, 5, 7, 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];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294381 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A293076, A293765.

Sequence in context: A262349 A109155 A338112 * A374654 A081072 A000717

Adjacent sequences: A294378 A294379 A294380 * A294382 A294383 A294384

Keyword

nonn,easy

Author

Clark Kimberling, Oct 29 2017

Extensions

More terms from Jack W Grahl, Apr 26 2018

Status

approved