A295949 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 8, 16, 31, 56, 97, 164, 273, 450, 737, 1202, 1956, 3176, 5151, 8347, 13519, 21888, 35430, 57342, 92797, 150165, 242989, 393182, 636200, 1029412, 1665644, 2695089, 4360767, 7055891, 11416694, 18472622, 29889354, 48362015, 78251409, 126613465, 204864916

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.

Formula

a(n) = H + R, where H = f(n-1)*a(0) + f(n)*a(1) and R = f(n-1)*b(2) + f(n-2)*b(3) + ... + f(2)*b(n-1) + f(1)*b(n), where f(n) = A000045(n), the n-th Fibonacci number.

Example

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

Mathematica

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n];
j = 1; While[j < 5, 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}]  (* A295949 *)
Table[b[n], {n, 0, 20}] (* complement *)

Crossrefs

Cf. A001622, A000045, A295862.

Sequence in context: A294534 A294542 A294553 * A077666 A232358 A212318

Adjacent sequences: A295946 A295947 A295948 * A295950 A295951 A295952

Keyword

nonn,easy

Author

Clark Kimberling, Dec 08 2017

Status

approved