A298006 Solution b( ) 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.

3, 4, 5, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 20, 21, 23, 25, 26, 27, 28, 30, 32, 33, 34, 35, 37, 39, 40, 41, 42, 44, 46, 47, 49, 50, 52, 53, 54, 55, 57, 58, 59, 61, 63, 64, 66, 67, 69, 70, 71, 72, 74, 75, 76, 78, 80, 81, 83, 84, 86, 87, 88, 89, 91, 92, 93

Offset: 0

Author

Clark Kimberling, Feb 09 2018

Keywords

• nonn
• easy

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. The solution a( ) is given at A297831. See A297830 for a guide to related sequences.

Conjecture: 9/10 < a(n) - n*sqrt(2) < 3 for n >= 1.

Links

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

Crossrefs

Cf. A297830, A297831.

Programs

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 < 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}]; (* A297831 *)
v = Table[b[n], {n, 0, k}]; (* A298006 *)
Take[u, 50]
Take[v, 50]