A296278 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n-1)*b(n), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 63, 185, 458, 979, 1941, 3640, 6571, 11531, 19818, 33533, 56081, 92974, 153135, 251005, 409954, 667799, 1085733, 1762772, 2859131, 4634047, 7506978, 12156625, 19681153, 31857434, 51560511, 83442305, 135029786, 218501851, 353564373, 572102128, 925705771

Offset

0,2

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) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5
a(2) = a(0) + a(1) + b(0)*b(1)*b(2) = 63
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...)

Mathematica

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[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}]; (* A296278 *)
Table[b[n], {n, 0, 20}]    (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A209183 A200802 A262005 * A239744 A024238 A234279

Adjacent sequences: A296275 A296276 A296277 * A296279 A296280 A296281

Keyword

nonn,easy

Author

Clark Kimberling, Dec 13 2017

Status

approved