A295955 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 1, where a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

3, 4, 13, 24, 45, 78, 133, 222, 367, 602, 984, 1602, 2603, 4223, 6845, 11088, 17954, 29064, 47041, 76129, 123196, 199352, 322576, 521957, 844563, 1366551, 2211146, 3577730, 5788910, 9366675, 15155621, 24522333, 39677992, 64200364, 103878396, 168078801

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

Mathematica

a[0] = 3; a[1] = 4; b[0] = 1; b[1] = 2; 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}];  (* A295955 *)

Crossrefs

Cf. A001622, A000045, A295862.

Sequence in context: A358583 A026700 A187775 * A151521 A142860 A111954

Adjacent sequences: A295952 A295953 A295954 * A295956 A295957 A295958

Keyword

nonn,easy

Author

Clark Kimberling, Dec 08 2017

Extensions

Definition corrected by Georg Fischer, Sep 27 2020

Status

approved