A295963 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 7, 14, 28, 50, 87, 147, 245, 404, 663, 1082, 1761, 2860, 4639, 7518, 12177, 19716, 31915, 51654, 83593, 135272, 218891, 354191, 573111, 927332, 1500474, 2427838, 3928345, 6356217, 10284597, 16640850, 26925484, 43566372, 70491895, 114058307, 184550243

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 A295862 for a guide to related sequences.

Links

Clark Kimberling, Table of n, a(n) for n = 0..2000

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
b(3) = 6 (least "new number")
a(2) = a(1) + a(0) + b(2) - 1 = 7
Complement: (b(n)) = (3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, ...)

Mathematica

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] - 1;
j = 1; While[j < 6, 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}];  (* A295963 *)

Crossrefs

Cf. A001622, A000045, A295862.

Sequence in context: A068040 A200084 A184704 * A221317 A005998 A122751

Adjacent sequences: A295960 A295961 A295962 * A295964 A295965 A295966

Keyword

nonn,easy

Author

Clark Kimberling, Dec 08 2017

Status

approved