A295953 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 1, 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, 10, 20, 38, 67, 115, 194, 322, 530, 867, 1413, 2297, 3728, 6044, 9792, 15858, 25673, 41555, 67253, 108834, 176114, 284976, 461119, 746125, 1207275, 1953432, 3160740, 5114206, 8274981, 13389223, 21664241, 35053502, 56717783, 91771326, 148489151

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) = 3, b(0) = 2, b(1) = 4, b(2) = 5
b(3) = 6 (least "new number")
a(2) = a(1) + a(0) + b(2) + 1 = 10
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, ...)

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] + 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}];  (* A295953 *)

Crossrefs

Cf. A001622, A000045, A295862.

Sequence in context: A338631 A272764 A293765 * A005997 A213850 A336529

Adjacent sequences: A295950 A295951 A295952 * A295954 A295955 A295956

Keyword

nonn,easy

Author

Clark Kimberling, Dec 08 2017

Status

approved