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

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

Offset

0,5

Comments

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.

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:

- 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 Fibonacci-like 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