login
A296849
Solution of the complementary equation a(n) = 2*a(n-1) + a(n-2) + b(n), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.
3
1, 2, 10, 28, 73, 182, 446, 1085, 2628, 6354, 15350, 37069, 89504, 216094, 521710, 1259533, 3040796, 7341146, 17723110, 42787389, 103297912, 249383238, 602064414, 1453512093, 3509088629, 8471689381, 20452467422, 49376624257, 119205715969, 287788056229
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, 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