A296291 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-1), where a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

2, 3, 13, 31, 68, 134, 250, 447, 777, 1323, 2220, 3697, 6097, 10002, 16337, 26609, 43250, 70199, 113827, 184444, 298731, 483679, 782960, 1267237, 2050845, 3318782, 5370381, 8689973, 14061250, 22752180, 36814450, 59567715, 96383317, 155952253, 252336862

Offset

0,1

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622). See A296245 for a guide to related sequences.

Links

Clark Kimberling, Table of n, a(n) for n = 0..1000

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

Example

a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, b(2) = 5
a(2) = a(0) + a(1) + 2*b(1) = 11
Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, ...)

Mathematica

a[0] = 2; a[1] = 3; b[0] = 1; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n-1];
j = 1; While[j < 10, k = a[j] - j - 1;
 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, k}]; (* A296291 *)
Table[b[n], {n, 0, 20}]    (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A092175 A317898 A317187 * A072997 A037428 A073688

Adjacent sequences: A296288 A296289 A296290 * A296292 A296293 A296294

Keyword

nonn,easy

Author

Clark Kimberling, Dec 14 2017

Status

approved