A296283 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n-1)*b(n), where a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

3, 4, 17, 81, 308, 725, 1537, 2982, 5509, 9811, 17036, 29031, 48797, 81188, 134305, 220965, 362110, 591055, 962405, 1564086, 2538635, 4116521, 6670756, 10804827, 17495239, 28321990, 45841589, 74190549, 120061898, 194285183, 314382985, 508707438, 823133263

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) = 3, a(1) = 4, b(0) = 1, b(1) = 2, b(2) = 5
a(2) = a(0) + a(1) + b(0)*b(1)*b(2) = 17
Complement: (b(n)) = (1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, ...)

Mathematica

a[0] = 3; a[1] = 4; b[0] = 1; b[1] = 2; 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}]; (* A296283 *)
Table[b[n], {n, 0, 20}]    (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A303480 A254200 A009208 * A051473 A039563 A032829

Adjacent sequences: A296280 A296281 A296282 * A296284 A296285 A296286

Keyword

nonn,easy

Author

Clark Kimberling, Dec 13 2017

Status

approved