A297835 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n + 1, 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, 10, 13, 16, 19, 22, 25, 30, 32, 37, 39, 44, 46, 51, 53, 58, 60, 65, 67, 70, 73, 78, 82, 84, 87, 90, 95, 99, 101, 104, 107, 112, 116, 118, 121, 124, 129, 133, 135, 138, 141, 146, 150, 152, 155, 158, 163, 167, 169, 174, 176, 181, 183, 186, 189, 194, 196

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: a(n) - (2 +sqrt(2))*n < 7 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) = 10.
Complement: (b(n)) = (3,4,6,7,8,9,11,12,14,15,17,18,20,...)

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] + 2 n + 1;
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}]  (* A297835 *)

Crossrefs

Cf. A297826, A297830.

Sequence in context: A343476 A343477 A296220 * A298000 A058216 A297998

Adjacent sequences: A297832 A297833 A297834 * A297836 A297837 A297838

Keyword

nonn,easy

Author

Clark Kimberling, Feb 04 2018

Status

approved