A296217 Solution of the complementary equation a(n) = a(1)*b(n-2) + a(2)*b(n-3) + ... + a(n-1)*b(0), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 6, 26, 112, 484, 2088, 9008, 38862, 167658, 723308, 3120486, 13462360, 58079138, 250564260, 1080981064, 4663554414, 20119445656, 86799050160, 374467330636, 1615522076050, 6969664279584, 30068434774274, 129720849313094, 559639996988064, 2414391579204576

Offset

0,2

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295862 for a guide to related sequences.

Links

Table of n, a(n) for n=0..25.

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

Example

a(0) =1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 4
a(2) = a(1)*b(0) = 2
Complement: (b(n)) = (3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...)

Mathematica

a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 1, n - 1}];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
u = Table[a[n], {n, 0, 200}];  (* A296217 *)
Table[b[n], {n, 0, 20}]
N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
RealDigits[Last[t], 10][[1]] (* A296218 *)

Crossrefs

Cf. A296000, A296218.

Sequence in context: A192403 A282618 A192435 * A050890 A114710 A230245

Adjacent sequences: A296214 A296215 A296216 * A296218 A296219 A296220

Keyword

nonn,easy

Author

Clark Kimberling, Dec 08 2017

Status

approved