A296259 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)^2, where a(0) = 2, a(1) = 3, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.

2, 3, 6, 25, 56, 130, 250, 461, 811, 1393, 2348, 3910, 6454, 10589, 17299, 28177, 45800, 74338, 120538, 195317, 316339, 512185, 829100, 1341961, 2171790, 3514535, 5687166, 9202601, 14890728, 24094353, 38986170, 63081679, 102069074, 165152049, 267222492

Offset

0,1

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(5))/2 = golden ratio (A001622). See A296245 for a guide to related sequences.

Links

Clark Kimberling, Table of n, a(n) for n = 0..1000

Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Formula

a(n) = H + R, where H = f(n-1)*a(0) + f(n)*a(1) and R = f(n-1)*b(0)^2 + f(n-2)*b(1)^2 + ... + f(2)*b(n-3)^2 + f(1)*b(n-2)^2, where f(n) = A000045(n), the n-th Fibonacci number.

Example

a(0) = 2, a(1) = 3, b(0) = 1;
a(2) = a(0) + a(1) + b(0)^2 = 6;
Complement: (b(n)) = (1, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...)

Mathematica

a[0] = 2; a[1] = 3; b[0] = 1;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n-2]^2;
j = 1; While[j < 6 , k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, k}]   (* A296259 *)
Table[b[n], {n, 0, 20}]  (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A032540 A063728 A333420 * A344935 A000341 A144857

Adjacent sequences: A296256 A296257 A296258 * A296260 A296261 A296262

Keyword

nonn,easy

Author

Clark Kimberling, Dec 11 2017

Status

approved