A295958 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 1, where a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

2, 3, 11, 21, 40, 70, 120, 201, 334, 549, 898, 1463, 2378, 3859, 6256, 10135, 16412, 26570, 43006, 69601, 112633, 182261, 294922, 477212, 772164, 1249407, 2021603, 3271043, 5292680, 8563758, 13856474, 22420269, 36276781, 58697089, 94973910, 153671041

Offset

0,1

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 = 11
Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, ...)

Mathematica

a[0] = 2; a[1] = 3; b[0] = 1; 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}];  (* A295958 *)

Crossrefs

Cf. A001622, A000045, A295862.

Sequence in context: A359017 A023182 A302310 * A049083 A305719 A002778

Adjacent sequences: A295955 A295956 A295957 * A295959 A295960 A295961

Keyword

nonn,easy

Author

Clark Kimberling, Dec 08 2017

Status

approved