A296258 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)^2, where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 3, 8, 27, 60, 123, 232, 436, 768, 1325, 2237, 3731, 6164, 10120, 16540, 26949, 43813, 71123, 115336, 186900, 302720, 490149, 793445, 1284219, 2078340, 3363343, 5442524, 8806767, 14250252, 23058043, 37309384, 60368583, 97679192, 158049071, 255729632

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

Mathematica

a[0] = 1; a[1] = 3; b[0] = 2;
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}]     (* A296258 *)
Table[b[n], {n, 0, 20}]  (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A023637 A128894 A118165 * A066020 A066018 A066023

Adjacent sequences: A296255 A296256 A296257 * A296259 A296260 A296261

Keyword

nonn,easy

Author

Clark Kimberling, Dec 11 2017

Status

approved