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A286327 Least possible sum of the squares of the two initial terms of a Fibonacci-like 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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 26 09:35 EST 2022. Contains 350598 sequences. (Running on oeis4.)