A296297 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*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, 16, 38, 82, 160, 296, 526, 910, 1544, 2584, 4282, 7046, 11549, 18847, 30681, 49848, 80886, 131130, 212453, 344063, 557041, 901676, 1459338, 2361686, 3821749, 6184215, 10006801, 16191912, 26199670, 42392602, 68593357, 110987111, 179581689, 290570126

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

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] + 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}]; (* A296297 *)
Table[b[n], {n, 0, 20}]    (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A000216 A110998 A051861 * A290265 A223093 A333022

Adjacent sequences: A296294 A296295 A296296 * A296298 A296299 A296300

Keyword

nonn,easy

Author

Clark Kimberling, Dec 14 2017

Status

approved