OFFSET
0,2
COMMENTS
The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. Conjecture: a(n) - (2 +sqrt(2))*n < 3 for n >= 1.
Guide to related sequences having initial values a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, where (b(n)) is the increasing sequence of positive integers not in (a(n)):
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a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n + 1 (a(n)) = A297835;
a(n) = a(1)*b(n-1) - a(0)*b(n-2)+floor(5*n/2) (a(n)) = A297998;
***
For sequences (a(n)) and (b(n)) associated with equations of the form a(n) = a(1)*b(n) - a(0)*b(n-1), see the guide at A297800.
LINKS
Clark Kimberling, Table of n, a(n) for n = 0..10000
EXAMPLE
a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 9.
Complement: (b(n)) = (3,4,5,6,8,10,11,13,14,16,17,19,...)
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;
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}] (* A297830 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 04 2018
STATUS
approved