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A305847 Solution a() of the complementary equation a(n) + b(n) = 5*n, where a(1) = 1. See Comments. 3
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 (list; graph; refs; listen; history; text; internal format)
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: A137217 A023705 A188070 * A248565 A065896 A099308

Adjacent sequences:  A305844 A305845 A305846 * A305848 A305849 A305850

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jun 11 2018

STATUS

approved

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Last modified May 6 09:25 EDT 2021. Contains 343580 sequences. (Running on oeis4.)