A296263 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n-1), where a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

2, 3, 9, 32, 71, 145, 272, 497, 879, 1508, 2543, 4233, 6986, 11459, 18717, 30482, 49541, 80403, 130364, 211229, 342099, 553880, 896579, 1451109, 2348390, 3800255, 6149457, 9950582, 16100969, 26052574, 42154665, 68208429, 110364354, 178574115, 288939875

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

Mathematica

a[0] = 2; a[1] = 3; b[0] = 1; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] b[n - 2];
j = 1; While[j < 10, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
u = Table[a[n], {n, 0, k}];  (* A296263 *)
Table[b[n], {n, 0, 20}]      (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A322752 A277345 A259943 * A064020 A204442 A306523

Adjacent sequences: A296260 A296261 A296262 * A296264 A296265 A296266

Keyword

nonn,easy

Author

Clark Kimberling, Dec 11 2017

Status

approved