A296294 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 3, 14, 35, 77, 152, 283, 505, 876, 1489, 2495, 4149, 6836, 11206, 18294, 29785, 48399, 78541, 127336, 206314, 334130, 540969, 875671, 1417261, 2293604, 3711590, 6005974, 9718401, 15725271, 25444629, 41170920, 66616665, 107788769, 174406688, 282196783

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(2) = 14
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, ...)

Mathematica

a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; 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}]; (* A296294 *)
Table[b[n], {n, 0, 20}]    (* complement *)

Crossrefs

Cf. A001622, A296245.

Sequence in context: A081377 A050934 A110427 * A128916 A130287 A167858

Adjacent sequences: A296291 A296292 A296293 * A296295 A296296 A296297

Keyword

nonn,easy

Author

Clark Kimberling, Dec 14 2017

Status

approved