A297834 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 4, 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, 5, 8, 12, 17, 19, 22, 27, 29, 32, 35, 40, 44, 46, 51, 53, 56, 59, 64, 68, 70, 75, 77, 82, 84, 87, 90, 95, 97, 100, 105, 109, 111, 114, 117, 122, 126, 128, 133, 135, 140, 142, 145, 148, 153, 155, 158, 163, 167, 169, 172, 175, 180, 184, 186, 189, 192

Offset

0,2

Comments

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

Conjecture: -3 < a(n) - (2 +sqrt(2))*n <= 1 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) = 5.
Complement: (b(n)) = (3,4,6,7,9,10,11,13,14,15,16,18,20,...)

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] + 2 n - 4;
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}]  (* A297834 *)

Crossrefs

Cf. A297826, A297830.

Sequence in context: A174605 A108577 A272719 * A036789 A330188 A214047

Adjacent sequences: A297831 A297832 A297833 * A297835 A297836 A297837

Keyword

nonn,easy

Author

Clark Kimberling, Feb 04 2018

Status

approved