A295947 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n), where a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

2, 4, 11, 21, 39, 68, 116, 194, 322, 529, 865, 1409, 2290, 3716, 6024, 9759, 15803, 25584, 41410, 67018, 108453, 175497, 283977, 459502, 743508, 1203040, 1946579, 3149651, 5096263, 8245948, 13342246, 21588230, 34930513, 56518781, 91449334, 147968156

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

Mathematica

a[0] = 1; a[1] = 3; b[0] = 2; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n];
j = 1; While[j < 16, 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}];  (* A295947 *)
Table[b[n], {n, 0, 20}]  (* complement *)

Crossrefs

Cf. A001622, A000045, A295862.

Sequence in context: A152597 A320679 A126972 * A018774 A102608 A356648

Adjacent sequences: A295944 A295945 A295946 * A295948 A295949 A295950

Keyword

nonn,easy

Author

Clark Kimberling, Dec 08 2017

Status

approved