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

%I A305847 #9 Jun 12 2018 10:21:29
%S A305847 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,
%T A305847 35,36,38,39,40,42,43,45,46,47,49,50,51,53,54,56,57,58,60,61,62,64,65,
%U A305847 67,68,69,71,72,74,75,76,78,79,80,82,83,85,86,87,89,90
%N A305847 Solution a() of the complementary equation a(n) + b(n) = 5*n, where a(1) = 1. See Comments.
%C A305847 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
%C A305847 2 - r <= n*x - a(n) < r and 2 - r < b(n) - n*y < r for all n >= 1.
%H A305847 Clark Kimberling, <a href="/A305847/b305847.txt">Table of n, a(n) for n = 1..10000</a>
%e A305847 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.
%t A305847 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A305847 u = 5; v = 5; z = 220;
%t A305847 c = {v}; a = {1}; b = {Last[c] - Last[a]};
%t A305847 Do[AppendTo[a, mex[Flatten[{a, b}], Last[a]]];
%t A305847   AppendTo[c, u Length[c] + v];
%t A305847   AppendTo[b, Last[c] - Last[a]], {z}];
%t A305847 c = Flatten[Position[Differences[a], 2]];
%t A305847 a  (* A305847 *)
%t A305847 b  (* A305848 *)
%t A305847 c  (* A305849 *)
%t A305847 (* _Peter J. C. Moses_, May 30 2018 *)
%Y A305847 Cf. A001622, A305848, A305849, A001614, A118011.
%K A305847 nonn,easy
%O A305847 1,2
%A A305847 _Clark Kimberling_, Jun 11 2018