A296251 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)^2, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 19, 46, 101, 196, 361, 638, 1099, 1858, 3101, 5128, 8425, 13778, 22459, 36526, 59309, 96235, 155985, 252704, 409218, 662498, 1072341, 1735515, 2808585, 4544884, 7354310, 11900094, 19255365, 31156483, 50412937, 81570576, 131984738, 213556610, 345542717

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(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(1)^2 + f(n-2)*b(2)^2 + ... + f(2)*b(n-2)^2 + f(1)*b(n-1)^2, where f(n) = A000045(n), the n-th Fibonacci number.

Example

a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4;
a(2) = a(0) + a(1) + b(1)^2 = 19;
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, ...)

Mathematica

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n-1]^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}]   (* A296251 *)
Table[b[n], {n, 0, 20}]  (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A062587 A109946 A141067 * A307554 A031911 A136685

Adjacent sequences: A296248 A296249 A296250 * A296252 A296253 A296254

Keyword

nonn,easy

Author

Clark Kimberling, Dec 10 2017

Status

approved