A305848 Solution b() of the complementary equation a(n) + b(n) = 5n, where a(1) = 1. See Comments.

4, 8, 12, 15, 19, 23, 26, 30, 34, 37, 41, 44, 48, 52, 55, 59, 63, 66, 70, 73, 77, 81, 84, 88, 92, 95, 99, 102, 106, 110, 113, 117, 120, 124, 128, 131, 135, 139, 142, 146, 149, 153, 157, 160, 164, 168, 171, 175, 178, 182, 186, 189, 193, 196, 200, 204, 207

Offset: 1

Author

Clark Kimberling, Jun 11 2018

Keywords

• nonn
• easy

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.

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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 *)