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

0,2

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 < 5/2 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) = 8.
Complement: (b(n)) = (3,4,5,6,7,9,10,12,13,15,16,18,19,...)

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 *)

Crossrefs

Cf. A297826, A297830.

Sequence in context: A216538 A077820 A360649 * A045086 A064105 A129516

Adjacent sequences: A297828 A297829 A297830 * A297832 A297833 A297834

Keyword

nonn,easy

Author

Clark Kimberling, Feb 04 2018

Status

approved