A296284 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 9, 23, 52, 105, 199, 360, 639, 1098, 1857, 3098, 5123, 8416, 13763, 22434, 36485, 59242, 96087, 155728, 252255, 408487, 661292, 1070377, 1732317, 2803394, 4536465, 7340669, 11878002, 19219599, 31098591, 50319244, 81418955, 131739387, 213159600

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) = 3, b(1) = 4, b(2) = 5
a(2) = a(0) + a(1) + 2*b(0) = 9
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, ...)

Mathematica

a[0] = 1; a[1] = 2; b[0] = 3;
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}]; (* A296284 *)
Table[b[n], {n, 0, 20}]    (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A062445 A009304 A154118 * A115185 A349189 A091107

Adjacent sequences: A296281 A296282 A296283 * A296285 A296286 A296287

Keyword

nonn,easy

Author

Clark Kimberling, Dec 13 2017

Status

approved