A295966 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 1, a(1) = 5, b(0) = 2, b(1) = 3, b(2) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 5, 9, 19, 34, 60, 103, 173, 287, 472, 772, 1258, 2045, 3319, 5381, 8719, 14120, 22860, 37002, 59885, 96911, 156821, 253758, 410606, 664392, 1075027, 1739449, 2814507, 4553988, 7368529, 11922552, 19291117, 31213706, 50504861, 81718606, 132223507, 213942154

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) = 2, a(1) = 3, b(0) = 1, b(1) = 4, b(2) = 5
b(3) = 6 (least "new number")
a(2) = a(1) + a(0) + b(2) - 1 = 9
Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, ...)

Mathematica

a[0] = 1; a[1] = 5; b[0] = 2; b[1] = 3; b[2] = 4;
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}];  (* A295966 *)

Crossrefs

Cf. A001622, A000045, A295862.

Sequence in context: A255652 A061202 A235799 * A060161 A282730 A082674

Adjacent sequences: A295963 A295964 A295965 * A295967 A295968 A295969

Keyword

nonn,easy

Author

Clark Kimberling, Dec 08 2017

Status

approved