A297837 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 4*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, 13, 18, 23, 28, 33, 38, 43, 48, 53, 60, 64, 69, 74, 81, 85, 90, 95, 102, 106, 111, 116, 123, 127, 132, 137, 144, 148, 153, 158, 165, 169, 174, 179, 186, 190, 195, 200, 207, 211, 216, 221, 228, 232, 237, 242, 247, 252, 259, 263, 268, 275, 279, 284, 289

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.

Conjecture: a(n) - (3 + sqrt(5))*n < 3 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) = 13.
Complement: (b(n)) = (3,4,5,6,7,8,9,10,11,12,14,15,16,17,19,20,...)

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] + 4 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, A297836.

Sequence in context: A128852 A191765 A063615 * A246358 A262688 A020585

Adjacent sequences: A297834 A297835 A297836 * A297838 A297839 A297840

Keyword

nonn,easy

Author

Clark Kimberling, Feb 04 2018

Status

approved