A295965 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, 9, 17, 32, 56, 97, 163, 271, 446, 730, 1190, 1935, 3142, 5095, 8256, 13371, 21648, 35041, 56712, 91777, 148514, 240317, 388858, 629203, 1018090, 1647323, 2665445, 4312801, 6978280, 11291116, 18269432, 29560585, 47830055, 77390679, 125220774, 202611494

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

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}];  (* A295965 *)

Crossrefs

Cf. A001622, A000045, A295862.

Sequence in context: A272057 A056658 A302164 * A034467 A234646 A065965

Adjacent sequences: A295962 A295963 A295964 * A295966 A295967 A295968

Keyword

nonn,easy

Author

Clark Kimberling, Dec 08 2017

Status

approved