A296296 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*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, 15, 36, 79, 155, 288, 513, 889, 1510, 2529, 4193, 6914, 11328, 18494, 30107, 48921, 79385, 128702, 208524, 337706, 546755, 885033, 1432409, 2318114, 3751248, 6070142, 9822227, 15893265, 25716449, 41610734, 67328268, 108940186, 176269708, 285211220

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

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] + n*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}]; (* A296296 *)
Table[b[n], {n, 0, 20}]    (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A060753 A241198 A356094 * A143880 A037388 A298370

Adjacent sequences: A296293 A296294 A296295 * A296297 A296298 A296299

Keyword

nonn,easy

Author

Clark Kimberling, Dec 14 2017

Status

approved