A305848 - OEIS

%I #8 Jun 12 2018 10:21:37

%S 4,8,12,15,19,23,26,30,34,37,41,44,48,52,55,59,63,66,70,73,77,81,84,

%T 88,92,95,99,102,106,110,113,117,120,124,128,131,135,139,142,146,149,

%U 153,157,160,164,168,171,175,178,182,186,189,193,196,200,204,207

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

%C 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 2 - r <= n*x - a(n) < r and 2 - r < b(n) - n*y < r for all n >= 1.

%H Clark Kimberling, <a href="/A305848/b305848.txt">Table of n, a(n) for n = 1..10000</a>

%e 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 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);

%t u = 5; v = 5; z = 220;

%t c = {v}; a = {1}; b = {Last[c] - Last[a]};

%t Do[AppendTo[a, mex[Flatten[{a, b}], Last[a]]];

%t   AppendTo[c, u Length[c] + v];

%t   AppendTo[b, Last[c] - Last[a]], {z}];

%t c = Flatten[Position[Differences[a], 2]];

%t a  (* A305847 *)

%t b  (* A305848 *)

%t c  (* A305849 *)

%t (* _Peter J. C. Moses_, May 30 2018 *)

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

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Jun 11 2018