A297836 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 3*n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments.

1, 2, 11, 15, 19, 23, 27, 31, 35, 41, 44, 48, 54, 57, 61, 67, 70, 74, 80, 83, 87, 93, 96, 100, 106, 109, 113, 119, 122, 126, 130, 134, 140, 143, 149, 152, 156, 162, 165, 169, 173, 177, 183, 186, 192, 195, 199, 205, 208, 212, 216, 220, 226, 229, 235, 238, 242

Offset

0,2

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. For a guide to related sequences, see A297830.

Conjectures: a(n) - (5 + sqrt(13))*n/2 < 2 for n >= 1.

Links

Clark Kimberling, Table of n, a(n) for n = 0..10000

Example

a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 11.
Complement: (b(n)) = (3,4,5,6,7,8,9,10,12,13,14,16,17,18,20,...)

Mathematica

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + 3 n;
j = 1; While[j < 100, k = a[j] - j - 1;
 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k
Table[a[n], {n, 0, k}]  (* A297836 *)

Crossrefs

Cf. A297826, A297830, A297837.

Sequence in context: A168498 A255370 A061845 * A241757 A272883 A257283

Adjacent sequences: A297833 A297834 A297835 * A297837 A297838 A297839

Keyword

nonn,easy

Author

Clark Kimberling, Feb 04 2018

Status

approved