A296288 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 11, 28, 63, 126, 237, 426, 743, 1277, 2150, 3581, 5911, 9700, 15849, 25819, 41972, 68131, 110481, 179030, 289971, 469505, 760026, 1230129, 1990803, 3221657, 5213240, 8435734, 13649870, 22086561, 35737451, 57825097, 93563700, 151390018, 244955010

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

Mathematica

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
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}]; (* A296288 *)
Table[b[n], {n, 0, 20}]    (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A161527 A143651 A054552 * A277361 A034534 A345035

Adjacent sequences: A296285 A296286 A296287 * A296289 A296290 A296291

Keyword

nonn,easy

Author

Clark Kimberling, Dec 14 2017

Status

approved