A296279 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) = 3, b(0) = 2, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 3, 44, 167, 421, 924, 1849, 3493, 6332, 11145, 19193, 32522, 54445, 90327, 148852, 244075, 398741, 649656, 1056377, 1715273, 2782276, 4509693, 7305769, 11831062, 19154381, 31005099, 50181404, 81210863, 131419237, 212659860, 344111833, 556807597, 900958700

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) = 3, b(0) = 2, b(1) = 4, b(2) = 5
a(2) = a(0) + a(1) + b(0)*b(1)*b(2) = 44
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...)

Mathematica

a[0] = 1; a[1] = 3; b[0] = 2; 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}]; (* A296279 *)
Table[b[n], {n, 0, 20}]    (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A076361 A130408 A349646 * A133073 A055539 A046946

Adjacent sequences: A296276 A296277 A296278 * A296280 A296281 A296282

Keyword

nonn,easy

Author

Clark Kimberling, Dec 13 2017

Status

approved