A298001 Solution of the complementary equation a(n) = a(1)*b(n) - a(0)*b(n-1) + 3*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, 12, 16, 20, 24, 28, 32, 36, 42, 45, 49, 55, 58, 62, 68, 71, 75, 81, 84, 88, 94, 97, 101, 107, 110, 114, 120, 123, 127, 131, 135, 141, 144, 150, 153, 157, 163, 166, 170, 174, 178, 184, 187, 193, 196, 200, 206, 209, 213, 217, 221, 227, 230, 236, 239, 243

Offset

0,2

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A298000 for a guide to related sequences.

Conjecture: a(n) - n*L < 4 for n >= 1, where L = (5 + sqrt(13))/2.

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, b(2) = 5, so that a(2) = 12.
Complement: (b(n)) = (3,4,5,6,8,9,10,11,14,15,17,18,19,21...)

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] + 3 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}]  (* A298001 *)

Crossrefs

Cf. A297800, A297826, A297836, A297837.

Sequence in context: A089021 A176888 A114974 * A071566 A295821 A113802

Adjacent sequences: A297998 A297999 A298000 * A298002 A298003 A298004

Keyword

nonn,easy

Author

Clark Kimberling, Feb 08 2018

Status

approved