A296257 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)^2, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 12, 30, 67, 133, 249, 446, 776, 1322, 2219, 3710, 6125, 10060, 16441, 26790, 43555, 70706, 114661, 185808, 300953, 487290, 788819, 1276734, 2066229, 3343692, 5410705, 8755238, 14166904, 22923166, 37091159, 60015481, 97107865, 157124642, 254233876

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

Mathematica

a[0] = 1; a[1] = 2; b[0] = 3;
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}]     (* A296257 *)
Table[b[n], {n, 0, 20}]  (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A249055 A127118 A259127 * A375623 A301774 A286230

Adjacent sequences: A296254 A296255 A296256 * A296258 A296259 A296260

Keyword

nonn,easy

Author

Clark Kimberling, Dec 11 2017

Status

approved