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

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]
Take[v, 50]

Crossrefs

Cf. A297830, A297834.

Sequence in context: A061208 A325439 A183860 * A184429 A248185 A357504

Adjacent sequences: A298106 A298107 A298108 * A298110 A298111 A298112

Keyword

nonn,easy

Author

Clark Kimberling, Feb 09 2018

Status

approved