A286321 Least possible strictly positive product of the two initial terms of a Fibonacci-like sequence containing n.

1, 1, 1, 2, 1, 4, 2, 1, 3, 4, 2, 6, 1, 3, 6, 4, 4, 2, 6, 5, 1, 8, 3, 9, 10, 4, 12, 4, 2, 15, 6, 9, 5, 1, 10, 8, 3, 6, 9, 20, 10, 4, 7, 12, 4, 12, 2, 8, 15, 6, 14, 16, 5, 18, 1, 16, 20, 8, 18, 3, 6, 19, 9, 24, 20, 10, 28, 4, 7, 30, 12, 32, 4, 12, 35, 2, 8, 14

Offset

1,4

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),

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

- A286326: 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 1 contains n).

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

For any n>2, a(A000032(n)) = 2.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, C program for A286321

Rémy Sigrist, Scatterplot of the first 100000 terms

Rémy Sigrist, Illustration of the first terms

Example

See illustration of the first terms in Links section.

Mathematica

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++]]; First@ DeleteCases[#, 0] &@ Union@ Map[Times @@ #[[1 ;; 2]] &, Select[s, MemberQ[#, n] &]]], {n, 78}] (* Michael De Vlieger, May 10 2017 *)

Crossrefs

Cf. A000032, A000045, A249783, A286326, A286327.

Sequence in context: A367647 A243070 A243060 * A349192 A118235 A346702

Adjacent sequences: A286318 A286319 A286320 * A286322 A286323 A286324

Keyword

nonn,look

Author

Rémy Sigrist, May 07 2017

Status

approved