OFFSET
0,2
COMMENTS
The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A298000 for a guide to related sequences.
Conjecture: a(n) - n*L < 4 for n >= 1, where L = (5 + sqrt(13))/2.
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, b(2) = 5, so that a(2) = 12.
Complement: (b(n)) = (3,4,5,6,8,9,10,11,14,15,17,18,19,21...)
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] + 3 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}] (* A298001 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 08 2018
STATUS
approved