OFFSET
0,2
COMMENTS
The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> 1 + sqrt(2) = A014176. [corrected by Clark Kimberling, Jun 09 2018]
LINKS
Clark Kimberling, Table of n, a(n) for n = 0..1000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
EXAMPLE
a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5
a(2) = 2*a(1) + a(0) + b(2) = 10
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, ...)
MATHEMATICA
a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
a[n_] := a[n] = 2*a[n - 1] + a[n - 2] + b[n];
j = 1; While[j < 7, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
u = Table[a[n], {n, 0, k}]; (* A296849 *)
Table[b[n], {n, 0, 20}] (* complement *)
Take[u, 30]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jan 12 2018
STATUS
approved