

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
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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



