A295952 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n), 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, 10, 21, 38, 67, 114, 192, 318, 523, 855, 1393, 2264, 3674, 5956, 9649, 15625, 25296, 40944, 66264, 107233, 173523, 280783, 454334, 735146, 1189510, 1924687, 3114229, 5038949, 8153212, 13192196, 21345444, 34537677, 55883160, 90420877, 146304078

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) = 5, b(0) = 2, b(1) = 3, b(2) = 4;
b(3) = 6 (least "new number");
a(2) = a(1) + a(0) + b(2) = 10;
Complement: (b(n)) = (2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, ...)

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];
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}]   (* A295952 *)
Table[b[n], {n, 0, 20}]  (* complement *)

Crossrefs

Cf. A001622, A000045, A295862.

Sequence in context: A002800 A280077 A372054 * A132174 A297301 A132461

Adjacent sequences: A295949 A295950 A295951 * A295953 A295954 A295955

Keyword

nonn,easy

Author

Clark Kimberling, Dec 08 2017

Status

approved