A296264 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) = 4, b(0) = 1, b(1) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

2, 4, 9, 28, 67, 137, 260, 477, 847, 1456, 2459, 4097, 6766, 11103, 18141, 29550, 48033, 77963, 126416, 204841, 331763, 537156, 869519, 1407325, 2277546, 3685654, 5964070, 9650654, 15615716, 25267426, 40884264, 66152880, 107038404, 173192616, 280232426

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

Mathematica

a[0] = 2; a[1] = 4; b[0] = 1; b[1] = 3;
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}]; (* A296264 *)
Table[b[n], {n, 0, 20}]     (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A349404 A332979 A176068 * A343166 A219665 A375632

Adjacent sequences: A296261 A296262 A296263 * A296265 A296266 A296267

Keyword

nonn,easy

Author

Clark Kimberling, Dec 12 2017

Status

approved