A286326 -id:A286326 - cp's OEIS frontend

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

Rémy Sigrist, May 07 2017

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).

			See illustration of the first terms in Links section.

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

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

{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 *)

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

Original entry on oeis.org

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

Rémy Sigrist, May 07 2017

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.

			See illustration of the first terms in Links section.

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

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

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 *)

A306696 Lexicographically earliest sequence of nonnegative terms such that for any n > 0 and k > 0, if a(n) >= a(n+k), then a(n+2*k) <> a(n+k).

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist, Mar 05 2019

nonn

This sequence has graphical features in common with A286326.

			For n=1:
- a(1) = 0 is suitable.
For n=2:
- a(2) = 0 is suitable.
For n=3:
- a(1) = 0 >= a(2) = 0, so a(3) <> 0,
- a(3) = 1 is suitable.
For n=4:
- a(2) = 0 < a(3) = 1,
- a(4) = 0 is suitable.
For n=5:
- a(3) = 1 >= a(4) = 0, so a(5) <> 0,
- a(1) = 0 < a(3) = 1,
- a(5) = 1 is suitable.
For n=6:
- a(4) = 0 < a(5) = 1,
- a(2) = 0 >= a(4) = 0, so a(6) <> 0,
- a(6) = 1 is suitable.
For n=7:
- a(5) = 1 >= a(6) = 1, so a(7) <> 1,
- a(3) = 1 >= a(5) = 1, so a(7) <> 1,
- a(1) = 0 >= a(4) = 0, so a(7) <> 0,
- a(7) = 2 is suitable.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C++ program for A306696
Rémy Sigrist, Colored scatterplot of (n, a(n)) for n = 1..100000 (where the color is function of the least k >= 0 such that (1+a(n))/n >= 2^k/A007583(k))

Cf. A000079, A007583, A164095, A181497, A286326.

Empirically:
- a(n) = 0 iff n is a power of 2 (A000079),
- a(n) = 1 iff n = 3 or belongs to A164095,
- a(2*n) = a(n),
- A181497(n) is the least k such that a(k) = n.

cp's OEIS Frontend

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

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

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Mathematica

A306696 Lexicographically earliest sequence of nonnegative terms such that for any n > 0 and k > 0, if a(n) >= a(n+k), then a(n+2*k) <> a(n+k).

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