A295951 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n), 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, 10, 19, 36, 63, 108, 182, 302, 497, 813, 1325, 2154, 3496, 5668, 9184, 14873, 24079, 38975, 63078, 102078, 165182, 267287, 432497, 699813, 1132340, 1832184, 2964556, 4796773, 7761363, 12558171, 20319571, 32877780, 53197390, 86075210, 139272641

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.

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) = 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) = 10;
Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17,  ...)

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];
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}]  (* A295951 *)
Table[b[n], {n, 0, 20}]  (* complement *)

Crossrefs

Cf. A001622, A000045, A295862.

Sequence in context: A175569 A275020 A122822 * A083944 A306106 A165550

Adjacent sequences: A295948 A295949 A295950 * A295952 A295953 A295954

Keyword

nonn,easy

Author

Clark Kimberling, Dec 08 2017

Status

approved