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A298109

Solution b( ) of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 4, 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, 6, 7, 9, 10, 11, 13, 14, 15, 16, 18, 20, 21, 23, 24, 25, 26, 28, 30, 31, 33, 34, 36, 37, 38, 39, 41, 42, 43, 45, 47, 48, 49, 50, 52, 54, 55, 57, 58, 60, 61, 62, 63, 65, 66, 67, 69, 71, 72, 73, 74, 76, 78, 79, 80, 81, 83, 85, 86, 88, 89, 91, 92, 93, 94

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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 - 4;

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}]; (* A297834 *)

v = Table[b[n], {n, 0, k}]; (* A298109 *)

Take[u, 50]