login
A297831
Solution 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
1, 2, 8, 11, 14, 17, 22, 24, 29, 31, 36, 38, 43, 45, 48, 51, 56, 60, 62, 65, 68, 73, 77, 79, 82, 85, 90, 94, 96, 99, 102, 107, 111, 113, 118, 120, 125, 127, 130, 133, 138, 140, 143, 148, 152, 154, 159, 161, 166, 168, 171, 174, 179, 181, 184, 189, 193, 195
OFFSET
0,2
COMMENTS
The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A297830 for a guide to related sequences.
Conjecture: a(n) - (2 +sqrt(2))*n < 5/2 for n >= 1.
LINKS
EXAMPLE
a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 8.
Complement: (b(n)) = (3,4,5,6,7,9,10,12,13,15,16,18,19,...)
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 < 100, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k
Table[a[n], {n, 0, k}] (* A297831 *)
CROSSREFS
Sequence in context: A216538 A077820 A360649 * A045086 A366915 A064105
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 04 2018
STATUS
approved