A296271 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n), where a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

3, 4, 12, 28, 75, 151, 289, 520, 908, 1558, 2620, 4373, 7217, 11845, 19350, 31518, 51228, 83145, 134813, 218441, 353782, 572798, 927204, 1500677, 2428635, 3930122, 6359656, 10290738, 16651417, 26943243, 43595815, 70540282, 114137392, 184679042, 298817877

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.

Example

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

Mathematica

a[0] = 3; a[1] = 4; b[0] = 1; b[1] = 2; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2] b[n];
j = 1; While[j < 10, 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}];  (* A296271 *)
Table[b[n], {n, 0, 20}]  (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A090660 A288140 A287594 * A000208 A079154 A101716

Adjacent sequences: A296268 A296269 A296270 * A296272 A296273 A296274

Keyword

nonn,easy

Author

Clark Kimberling, Dec 12 2017

Status

approved