A295959 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, 11, 23, 42, 74, 126, 211, 350, 575, 940, 1531, 2488, 4037, 6544, 10601, 17166, 27789, 44978, 72792, 117796, 190615, 308439, 499083, 807552, 1306666, 2114250, 3420949, 5535233, 8956217, 14491486, 23447740, 37939264, 61387043, 99326347, 160713431

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) = 5, b(0) = 2, b(1) = 3, b(2) = 4
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] = 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}];  (* A295959 *)

Crossrefs

Cf. A001622, A000045, A295862.

Sequence in context: A343711 A323042 A210731 * A156109 A192954 A337437

Adjacent sequences: A295956 A295957 A295958 * A295960 A295961 A295962

Keyword

nonn,easy

Author

Clark Kimberling, Dec 08 2017

Status

approved