login
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.
7
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
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
Sequence in context: A262349 A109155 A338112 * A374654 A081072 A000717
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Oct 29 2017
EXTENSIONS
More terms from Jack W Grahl, Apr 26 2018
STATUS
approved