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%I #23 Mar 09 2019 05:11:37
%S 0,1,1,1,2,1,2,2,1,3,2,2,3,1,3,3,2,4,2,3,4,1,4,3,3,5,2,4,4,2,5,3,4,5,
%T 1,5,4,3,6,3,5,5,2,6,4,4,6,2,6,5,3,7,4,5,6,1,7,5,4,7,3,6,6,3,8,5,5,7,
%U 2,7,6,4,8,4,6,7,2,8,6,5,8,3,7,7,4,9,5
%N Least possible maximum of the two initial terms of a Fibonacci-like sequence containing n.
%C A Fibonacci-like sequence f satisfies f(n+2) = f(n+1) + f(n), and is uniquely identified by its two initial terms f(0) and f(1); here we consider Fibonacci-like sequences with f(0) >= 0 and f(1) >= 0.
%C This sequence is part of a family of variations of A249783, where we minimize a function g of the initial terms of Fibonacci-like sequences containing n:
%C - A249783: g(f) = f(0) + f(1),
%C - A286321: g(f) = f(0) * f(1),
%C - a: g(f) = max(f(0), f(1)),
%C - A286327: g(f) = f(0)^2 + f(1)^2.
%C For any n>0, a(n) <= n (as the Fibonacci-like sequence with initial terms n and 0 contains n).
%C For any n>0, a(A000045(n)) = 1.
%C Apparently the same as A097368 for n > 1. - _Georg Fischer_, Oct 09 2018
%H Rémy Sigrist, <a href="/A286326/b286326.txt">Table of n, a(n) for n = 0..10000</a>
%H Rémy Sigrist, <a href="/A286326/a286326.png">Scatterplot of the first 100000 terms</a>
%H Rémy Sigrist, <a href="/A286326/a286326.pdf">Illustration of the first terms</a>
%H Rémy Sigrist, <a href="/A286326/a286326_1.txt">C program for A286326</a>
%H Rémy Sigrist, <a href="/A286326/a286326_1.png">Colored scatterplot of (n, a(n)) for n = 1..100000</a> (where the color is function of the least k > 0 such that a(n)/n >= A000045(k)/A001654(k))
%e See illustration of the first terms in Links section.
%t {0}~Join~Table[Module[{a = 0, b = 1, s = {}}, While[a <= n, AppendTo[s, Flatten@ NestWhileList[{#2, #1 + #2} & @@ # &, {a, b}, Last@ # < n &]]; If[a + b >= n, a++; b = 1, b++]]; Min@ Map[Max@ #[[1 ;; 2]] &, Select[s, MemberQ[#, n] &]]], {n, 86}] (* _Michael De Vlieger_, May 10 2017 *)
%Y Cf. A000045, A001654, A097368, A249783, A286321, A286327.
%K nonn,look
%O 0,5
%A _Rémy Sigrist_, May 07 2017
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