A296292 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-1), where a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

2, 4, 12, 31, 67, 133, 248, 444, 772, 1315, 2217, 3686, 6083, 9977, 16298, 26545, 43147, 70032, 113557, 184007, 298024, 482535, 781109, 1264242, 2045999, 3310941, 5357694, 8669445, 14028035, 22698437, 36727492, 59427014, 96155658, 155583893, 251740843

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(1) = 12
Complement: (b(n)) = (1, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, ...)

Mathematica

a[0] = 2; a[1] = 4; b[0] = 1; b[1] = 3;
a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n-1];
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}]; (* A296292 *)
Table[b[n], {n, 0, 20}]    (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A148189 A148190 A151434 * A287966 A148191 A141312

Adjacent sequences: A296289 A296290 A296291 * A296293 A296294 A296295

Keyword

nonn,easy

Author

Clark Kimberling, Dec 14 2017

Status

approved