A298108 - OEIS

The OEIS is supported by the many generous donors to the OEIS Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give

A298108

Solution b( ) of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 3, 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.

3, 4, 5, 7, 8, 10, 11, 12, 13, 15, 17, 18, 19, 20, 22, 24, 25, 27, 28, 30, 31, 32, 33, 35, 36, 37, 39, 41, 42, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 58, 59, 60, 61, 63, 65, 66, 67, 68, 70, 72, 73, 75, 76, 78, 79, 80, 81, 83, 85, 86, 88, 89, 90, 91, 93, 94

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. The solution a( ) is given at A297832. See A297830 for a guide to related sequences.

Conjecture: 3/2 < a(n) - n*sqrt(2) < 4 for n >= 1.

LINKS

Table of n, a(n) for n=0..65.

Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

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 - 3;

j = 1; While[j < 80000, k = a[j] - j - 1;

While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k

u = Table[a[n], {n, 0, k}]; (* A297833 *)

v = Table[b[n], {n, 0, k}]; (* A298108 *)

Take[u, 50]