A296275 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*b(n), where a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

2, 3, 25, 58, 125, 239, 436, 765, 1311, 2208, 3675, 6065, 9950, 16255, 26477, 43038, 69857, 113275, 183552, 297289, 481347, 779188, 1261159, 2041049, 3302964, 5344825, 8648659, 13994414, 22644065, 36639535, 59284722, 95925447, 155211429, 251138208, 406351043

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) = 1, b(1) = 4, b(2) = 5;
a(2) = a(0) + a(1) + b(1)*b(2) = 25;
Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...)

Mathematica

a[0] = 2; a[1] = 3; b[0] = 1; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] 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}];  (* A296275 *)
Table[b[n], {n, 0, 20}]  (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A307922 A048674 A320223 * A295328 A281169 A296281

Adjacent sequences: A296272 A296273 A296274 * A296276 A296277 A296278

Keyword

nonn,easy

Author

Clark Kimberling, Dec 12 2017

Status

approved