A298003 Solution b( ) of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n, 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, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 24, 25, 27, 29, 30, 31, 32, 34, 36, 37, 38, 39, 41, 43, 44, 45, 46, 48, 50, 51, 52, 53, 55, 57, 58, 60, 61, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 77, 78, 80, 81, 82, 83, 85, 86, 87, 89, 91, 92, 94

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 A297830, which includes a guide to related sequences.

Conjecture: 3/5 < a(n) - n*sqrt(2)*n < 3 for n >= 1.

Links

Clark Kimberling, Table of n, a(n) for n = 0..10000

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;
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}]; (* A297830 *)
v = Table[b[n], {n, 0, k}]; (* A298003 *)
Take[u, 50]
Take[v, 50]

Crossrefs

Cf. A297830.

Sequence in context: A304808 A039180 A039134 * A377989 A207966 A097901

Adjacent sequences: A298000 A298001 A298002 * A298004 A298005 A298006

Keyword

nonn,easy

Author

Clark Kimberling, Feb 08 2018

Status

approved