A298000 Solution of the complementary equation a(n) = a(1)*b(n) - a(0)*b(n-1) + 2*n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments.

1, 2, 10, 13, 16, 19, 22, 27, 29, 34, 36, 41, 43, 48, 50, 55, 57, 60, 63, 68, 72, 74, 77, 80, 85, 89, 91, 94, 97, 102, 106, 108, 111, 114, 119, 123, 125, 128, 131, 136, 140, 142, 147, 149, 154, 156, 159, 162, 167, 169, 172, 177, 181, 183, 188, 190, 195, 197

Offset

0,2

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values.

Conjectures: a(n) - (2 +sqrt(2))*n < 4 for n >= 1. Guide to related sequences having initial values a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, where (b(n)) is the increasing sequence of positive integers not in (a(n)):

***

a(n) = a(1)*b(n) - a(0)*b(n-1) + n (a(n)) = A297999; (b(n)) = A298110

a(n) = a(1)*b(n) - a(0)*b(n-1) + 2*n (a(n)) = A298000; (b(n)) = A298111

a(n) = a(1)*b(n) - a(0)*b(n-1) + 3*n (a(n)) = A298001; (b(n)) = A298112

a(n) = a(1)*b(n) - a(0)*b(n-1) + 4*n (a(n)) = A298002; (b(n)) = A298113

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)) = (3,4,5,6,8,9,11,12,14,15,17,18,20,...)

Mathematica

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[1]*b[n] - a[0]*b[n - 1] + 2 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}]  (* A298000 *)

Crossrefs

Cf. A297826, A297836, A297837.

Sequence in context: A343477 A296220 A297835 * A058216 A297998 A037386

Adjacent sequences: A297997 A297998 A297999 * A298001 A298002 A298003

Keyword

nonn,easy

Author

Clark Kimberling, Feb 04 2018

Status

approved