A297831 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, 8, 11, 14, 17, 22, 24, 29, 31, 36, 38, 43, 45, 48, 51, 56, 60, 62, 65, 68, 73, 77, 79, 82, 85, 90, 94, 96, 99, 102, 107, 111, 113, 118, 120, 125, 127, 130, 133, 138, 140, 143, 148, 152, 154, 159, 161, 166, 168, 171, 174, 179, 181, 184, 189, 193, 195

Offset: 0

Author

Clark Kimberling, Feb 04 2018

Keywords

• nonn
• easy

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 < 5/2 for n >= 1.

Examples

			a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 8.
Complement: (b(n)) = (3,4,5,6,7,9,10,12,13,15,16,18,19,...)
		

Links

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

Crossrefs

Cf. A297826, A297830.

Programs

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}]  (* A297831 *)