A296286 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 3, 8, 23, 51, 104, 197, 364, 641, 1104, 1865, 3112, 5145, 8452, 13821, 22528, 36637, 59488, 96485, 156372, 253297, 410173, 664020, 1074791, 1739459, 2814950, 4555163, 7370923, 11926954, 19298805, 31226749, 50526608, 81754477, 132282273, 214038008

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

Mathematica

a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n-2];
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}]; (* A296286 *)
Table[b[n], {n, 0, 20}]    (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A183930 A183922 A340493 * A068602 A255834 A027212

Adjacent sequences: A296283 A296284 A296285 * A296287 A296288 A296289

Keyword

nonn,easy

Author

Clark Kimberling, Dec 14 2017

Status

approved