A286326 Least possible maximum of the two initial terms of a Fibonacci-like sequence containing n.

0, 1, 1, 1, 2, 1, 2, 2, 1, 3, 2, 2, 3, 1, 3, 3, 2, 4, 2, 3, 4, 1, 4, 3, 3, 5, 2, 4, 4, 2, 5, 3, 4, 5, 1, 5, 4, 3, 6, 3, 5, 5, 2, 6, 4, 4, 6, 2, 6, 5, 3, 7, 4, 5, 6, 1, 7, 5, 4, 7, 3, 6, 6, 3, 8, 5, 5, 7, 2, 7, 6, 4, 8, 4, 6, 7, 2, 8, 6, 5, 8, 3, 7, 7, 4, 9, 5

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),

- a: g(f) = max(f(0), f(1)),

- A286327: g(f) = f(0)^2 + f(1)^2.

For any n>0, a(n) <= n (as the Fibonacci-like sequence with initial terms n and 0 contains n).

For any n>0, a(A000045(n)) = 1.

Apparently the same as A097368 for n > 1. - Georg Fischer, Oct 09 2018

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 A286326

Rémy Sigrist, Colored scatterplot of (n, a(n)) for n = 1..100000 (where the color is function of the least k > 0 such that a(n)/n >= A000045(k)/A001654(k))

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[Max@ #[[1 ;; 2]] &, Select[s, MemberQ[#, n] &]]], {n, 86}] (* Michael De Vlieger, May 10 2017 *)

Crossrefs

Cf. A000045, A001654, A097368, A249783, A286321, A286327.

Sequence in context: A109967 A359539 A000119 * A097368 A271900 A194083

Adjacent sequences: A286323 A286324 A286325 * A286327 A286328 A286329

Keyword

nonn,look

Author

Rémy Sigrist, May 07 2017

Status

approved