A295950 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n), where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 4, 10, 20, 37, 65, 111, 187, 310, 510, 834, 1359, 2209, 3585, 5812, 9416, 15249, 24687, 39959, 64670, 104654, 169350, 274031, 443409, 717469, 1160908, 1878408, 3039348, 4917789, 7957171, 12874995, 20832202, 33707235, 54539476, 88246751, 142786268

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

Mathematica

a[0] = 1; a[1] = 4; b[0] = 2; b[1] = 3; 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}]   (* A295950 *)
Table[b[n], {n, 0, 20}]  (* complement *)

Crossrefs

Cf. A001622, A000045, A295862.

Sequence in context: A008058 A301170 A188280 * A038420 A008254 A301178

Adjacent sequences: A295947 A295948 A295949 * A295951 A295952 A295953

Keyword

nonn,easy

Author

Clark Kimberling, Dec 08 2017

Status

approved