A296276 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*b(n), where a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

2, 4, 21, 55, 118, 229, 419, 738, 1267, 2137, 3560, 5879, 9649, 15768, 25689, 41763, 67794, 109937, 178171, 288614, 467337, 756551, 1224538, 1981791, 3207085, 5189688, 8397643, 13588261, 21986896, 35576213, 57564231, 93141634, 150707125, 243850091, 394558622

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) = 1, b(1) = 3, b(2) = 5
a(2) = a(0) + a(1) + b(1)*b(2) = 21
Complement: (b(n)) = (1, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, ...)

Mathematica

a[0] = 2; a[1] = 4; b[0] = 1; b[1] = 3; 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}]; (* A296276 *)
Table[b[n], {n, 0, 20}]    (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A058522 A292534 A122736 * A092458 A071779 A107388

Adjacent sequences: A296273 A296274 A296275 * A296277 A296278 A296279

Keyword

nonn,easy

Author

Clark Kimberling, Dec 13 2017

Status

approved