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

Author

Rémy Sigrist, May 07 2017

Keywords

• nonn
• look

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).

Examples

			See illustration of the first terms in Links section.
		

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

Crossrefs

Cf. A000045, A001481, A081324, A249783, A286321, A286326.

Programs

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 *)