A286327 -id:A286327 - cp's OEIS frontend

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

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),
- 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

			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 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))

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

{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 *)

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 *)

A327194 For any n >= 0: consider the different ways to split the binary representation of n into two (possibly empty) parts, say with value x and y; a(n) is the least possible value of x^2 + y^2.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 5, 10, 1, 2, 5, 10, 9, 10, 13, 18, 1, 2, 5, 10, 17, 26, 29, 34, 9, 10, 13, 18, 25, 34, 45, 58, 1, 2, 5, 10, 17, 26, 37, 50, 25, 26, 29, 34, 41, 50, 61, 74, 9, 10, 13, 18, 25, 34, 45, 58, 49, 50, 53, 58, 65, 74, 85, 98, 1, 2, 5, 10, 17, 26, 37

Offset: 0

Rémy Sigrist, Aug 25 2019

This sequence shares graphical features with A286327.

			For n=42:
- the binary representation of 42 is "101010",
- there are 7 ways to split it:
   - "" and "101010": x=0 and y=42: 0^2 + 42^2 = 1764,
   - "1" and "01010": x=1 and y=10: 1^2 + 10^2 = 101,
   - "10" and "1010": x=2 and y=10: 2^2 + 10^2 = 104,
   - "101" and "010": x=5 and y=2: 5^2 + 2^2 = 29,
   - "1010" and "10": x=10 and y=2: 10^2 + 2^2 = 104,
   - "10101" and "0": x=21 and y=0: 21^2 + 0^2 = 441,
   - "101010" and "": x=42 and y=0: 42^2 + 0^2 = 1764,
- hence a(42) = 29.

Rémy Sigrist, Table of n, a(n) for n = 0..8192

See A327186 for other variants.

Cf. A286327.

Table[Min[Total[#^2]&/@Table[FromDigits[#,2]&/@TakeDrop[IntegerDigits[n,2],d],{d,0,IntegerLength[n,2]}]],{n,0,80}] (* Harvey P. Dale, Mar 03 2023 *)

a(n) = my (v=oo, b=binary(n)); for (w=0, #b, v=min(v, fromdigits(b[1..w],2)^2 + fromdigits(b[w+1..#b],2)^2)); v

a(n) = 1 iff n is a power of 2.

A341474 Let T be the set of sequences {t(k), k >= 0} such that for any k >= 3, t(k) = t(k-1) + t(k-2) + t(k-3); a(n) is the least possible value of t(0)^2 + t(1)^2 + t(2)^2 for an element t of T containing n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 4, 3, 2, 1, 4, 1, 4, 5, 5, 3, 2, 5, 1, 6, 4, 6, 1, 5, 4, 5, 9, 5, 9, 3, 10, 2, 10, 5, 8, 1, 6, 9, 4, 17, 6, 13, 1, 11, 5, 13, 4, 9, 5, 9, 16, 5, 18, 9, 14, 3, 14, 10, 9, 2, 12, 10, 5, 21, 8, 19, 1, 17, 6, 19, 9, 10, 4, 17, 17, 6, 26, 13

Offset: 0

Rémy Sigrist, Feb 13 2021

This sequence is a variant of A286327; here we consider tribonacci-like sequences, there Fibonacci like sequences. The scatterplots of these sequences are similar.

			The first terms of the elements t of T such that t(0)^2 + t(1)^2 + t(2)^2 <= 4 are:
  t(0)^2+t(1)^2+t(3)^2  t(0)  t(1)  t(2)  t(3)  t(4)  t(5)  t(6)  t(7)  t(8)  t(9)
  --------------------  ----  ----  ----  ----  ----  ----  ----  ----  ----  ----
                     0     0     0     0     0     0     0     0     0     0     0
                     1     0     0     1     1     2     4     7    13    24    44
                     1     0     1     0     1     2     3     6    11    20    37
                     1     1     0     0     1     1     2     4     7    13    24
                     2     0     1     1     2     4     7    13    24    44    81
                     2     1     0     1     2     3     6    11    20    37    68
                     2     1     1     0     2     3     5    10    18    33    61
                     3     1     1     1     3     5     9    17    31    57   105
                     4     0     0     2     2     4     8    14    26    48    88
                     4     0     2     0     2     4     6    12    22    40    74
                     4     2     0     0     2     2     4     8    14    26    48
- so a(0) = 0,
     a(1) = a(2) = a(3) = a(4) = a(6) = a(7) = a(11) = 1,
     a(5) = a(10) = a(18) = 2,
     a(9) = a(17) = 3,
     a(8) = a(12) = a(14) = 4.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Scatterplot of the first 100000 terms
Rémy Sigrist, Scatterplot of the first 1000000 terms
Rémy Sigrist, PARI program for A341474

Cf. A000073, A001590, A213816, A286327, A341456.

PARI
```
See Links section.
```

a(n) = 0 iff n = 0.
a(n) = 1 iff n belongs to A213816.
a(n) <= n^2.

cp's OEIS Frontend

A286326 Least possible maximum 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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A327194 For any n >= 0: consider the different ways to split the binary representation of n into two (possibly empty) parts, say with value x and y; a(n) is the least possible value of x^2 + y^2.

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A341474 Let T be the set of sequences {t(k), k >= 0} such that for any k >= 3, t(k) = t(k-1) + t(k-2) + t(k-3); a(n) is the least possible value of t(0)^2 + t(1)^2 + t(2)^2 for an element t of T containing n.

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Examples

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PARI

Formula