A305847 Solution a() of the complementary equation a(n) + b(n) = 5*n, where a(1) = 1. See Comments.

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 16, 17, 18, 20, 21, 22, 24, 25, 27, 28, 29, 31, 32, 33, 35, 36, 38, 39, 40, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 56, 57, 58, 60, 61, 62, 64, 65, 67, 68, 69, 71, 72, 74, 75, 76, 78, 79, 80, 82, 83, 85, 86, 87, 89, 90

Offset

1,2

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial value. Let x = (5 - sqrt(5))/2 and y = (5 + sqrt(5))/2. Let r = y - 2 = golden ratio (A001622). It appears that

2 - r <= n*x - a(n) < r and 2 - r < b(n) - n*y < r for all n >= 1.

Links

Clark Kimberling, Table of n, a(n) for n = 1..10000

Example

a(1) = 1, so b(1) = 5 - a(1) = 4.  In order for a() and b() to be increasing and complementary, we have a(2) = 2, a(3) = 3, a(4) = 5, etc.

Mathematica

mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
u = 5; v = 5; z = 220;
c = {v}; a = {1}; b = {Last[c] - Last[a]};
Do[AppendTo[a, mex[Flatten[{a, b}], Last[a]]];
  AppendTo[c, u Length[c] + v];
  AppendTo[b, Last[c] - Last[a]], {z}];
c = Flatten[Position[Differences[a], 2]];
a  (* A305847 *)
b  (* A305848 *)
c  (* A305849 *)
(* Peter J. C. Moses, May 30 2018 *)

Crossrefs

Cf. A001622, A305848, A305849, A001614, A118011.

Sequence in context: A367625 A023705 A188070 * A376958 A248565 A065896

Adjacent sequences: A305844 A305845 A305846 * A305848 A305849 A305850

Keyword

nonn,easy

Author

Clark Kimberling, Jun 11 2018

Status

approved