

A286327


Least possible sum of the squares of the two initial terms of a Fibonaccilike sequence containing n.


5



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, 5, 34, 13, 16, 26, 1, 41, 20, 10, 37, 9, 25, 29, 4, 50, 25, 17, 40, 5, 36, 34, 13, 53, 16, 26, 45, 1, 49, 41, 20, 58, 10, 37, 52, 9, 64, 25, 29, 65, 4, 50, 61, 25
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OFFSET

0,5


COMMENTS

A Fibonaccilike 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 Fibonaccilike sequences with f(0) >= 0 and f(1) >= 0.
This sequence is part of a family of variations of A249783, where we minimize a function g of the initial terms of Fibonaccilike sequences containing n:
 A249783: g(f) = f(0) + f(1),
 A286321: g(f) = f(0) * f(1),
 A286326: g(f) = max(f(0), f(1)),
 a: g(f) = f(0)^2 + f(1)^2.
For any n>0, a(n) <= n^2 (as the Fibonaccilike sequence with initial terms n and 0 contains n).
For any n>0, a(A000045(n)) = 1.
All terms belong to A001481 (numbers that are the sum of 2 squares).
No term > 0 belongs to A081324 (twice a square but not the sum of 2 distinct squares).


LINKS

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Scatterplot of the first 100000 terms
Rémy Sigrist, Illustration of the first terms
Rémy Sigrist, C program for A286327


EXAMPLE

See illustration of the first terms in Links section.


MATHEMATICA

{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 *)


CROSSREFS

Cf. A000045, A001481, A081324, A249783, A286321, A286326.
Sequence in context: A258853 A275791 A093561 * A081773 A324836 A302151
Adjacent sequences: A286324 A286325 A286326 * A286328 A286329 A286330


KEYWORD

nonn,look


AUTHOR

Rémy Sigrist, May 07 2017


STATUS

approved



