A298111 Solution b( ) of the complementary equation a(n) = a(1)*b(n) - a(0)*b(n-1) + 2*n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments.

3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 25, 26, 28, 30, 31, 32, 33, 35, 37, 38, 39, 40, 42, 44, 45, 46, 47, 49, 51, 52, 53, 54, 56, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 75, 76, 78, 79, 81, 82, 83, 84, 86, 87, 88, 90, 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 A298000. See A297830 for a guide to related sequences.

Conjecture: 1/5 < 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; b[2] = 5;
a[n_] := a[n] = a[1]*b[n] - a[0]*b[n - 1] + 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}]; (* A298000 *)
v = Table[b[n], {n, 0, k}]; (* A298111 *)
Take[u, 50]
Take[v, 50]

Crossrefs

Cf. A297830, A298000.

Sequence in context: A258187 A039237 A039181 * A299534 A026454 A026458

Adjacent sequences: A298108 A298109 A298110 * A298112 A298113 A298114

Keyword

nonn,easy

Author

Clark Kimberling, Feb 09 2018

Status

approved