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

0,1

Comments

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

Conjecture: 9/10 < a(n) - n*sqrt(2) < 3 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 - 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]

Crossrefs

Cf. A297830, A297831.

Sequence in context: A227938 A039096 A083121 * A026438 A026442 A334763

Adjacent sequences: A298003 A298004 A298005 * A298007 A298008 A298009

Keyword

nonn,easy

Author

Clark Kimberling, Feb 09 2018

Status

approved