A305848 -id:A305848 - cp's OEIS frontend

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

Clark Kimberling, Jun 11 2018

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.

			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.

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

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

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 *)

A305849 Positions of 2 in the difference sequence of A305847.

Original entry on oeis.org

3, 6, 9, 11, 14, 17, 19, 22, 25, 27, 30, 32, 35, 38, 40, 43, 46, 48, 51, 53, 56, 59, 61, 64, 67, 69, 72, 74, 77, 80, 82, 85, 87, 90, 93, 95, 98, 101, 103, 106, 108, 111, 114, 116, 119, 122, 124, 127, 129, 132, 135, 137, 140, 142, 145, 148, 150, 153, 156, 158

Offset: 1

Clark Kimberling, Jun 11 2018

This is also the sequence of positions of 3 in the difference sequence of A305848.

			A305837 = (1,2,3,5,6,7,9,10,11,13,14,16, ...);
differences: (1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, ...);
positions of 2: (3,6,9,11,14, ...).

Clark Kimberling, Table of n, a(n) for n = 1..3000

Cf. A001622, A305847, A305848.

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 *)

cp's OEIS Frontend

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

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