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A286327 Least possible sum of the squares of the two initial terms of a Fibonacci-like sequence containing n.

%I A286327 #23 May 10 2017 23:55:18
%S A286327 0,1,1,1,4,1,4,5,1,9,4,5,13,1,10,9,4,17,5,13,16,1,20,10,9,25,4,25,17,
%T A286327 5,34,13,16,26,1,41,20,10,37,9,25,29,4,50,25,17,40,5,36,34,13,53,16,
%U A286327 26,45,1,49,41,20,58,10,37,52,9,64,25,29,65,4,50,61,25
%N A286327 Least possible sum of the squares of the two initial terms of a Fibonacci-like sequence containing n.
%C A286327 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 A286327 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 A286327 - A249783: g(f) = f(0) + f(1),
%C A286327 - A286321: g(f) = f(0) * f(1),
%C A286327 - A286326: g(f) = max(f(0), f(1)),
%C A286327 - a:       g(f) = f(0)^2 + f(1)^2.
%C A286327 For any n>0, a(n) <= n^2 (as the Fibonacci-like sequence with initial terms n and 0 contains n).
%C A286327 For any n>0, a(A000045(n)) = 1.
%C A286327 All terms belong to A001481 (numbers that are the sum of 2 squares).
%C A286327 No term > 0 belongs to A081324 (twice a square but not the sum of 2 distinct squares).
%H A286327 Rémy Sigrist, <a href="/A286327/b286327.txt">Table of n, a(n) for n = 0..10000</a>
%H A286327 Rémy Sigrist, <a href="/A286327/a286327_1.png">Scatterplot of the first 100000 terms</a>
%H A286327 Rémy Sigrist, <a href="/A286327/a286327.pdf">Illustration of the first terms</a>
%H A286327 Rémy Sigrist, <a href="/A286327/a286327_3.txt">C program for A286327</a>
%e A286327 See illustration of the first terms in Links section.
%t A286327 {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[Total[(#[[1 ;; 2]])^2] &, Select[s, MemberQ[#, n] &]]], {n, 71}] (* _Michael De Vlieger_, May 10 2017 *)
%Y A286327 Cf. A000045, A001481, A081324, A249783, A286321, A286326.
%K A286327 nonn,look
%O A286327 0,5
%A A286327 _Rémy Sigrist_, May 07 2017