A249783 -id:A249783 - cp's OEIS frontend

A246399 Least k such that A249783(k) = n.

1, 4, 7, 12, 17, 22, 27, 43, 51, 59, 67, 75, 83, 122, 135, 148, 161, 174, 187, 200, 213, 226, 239, 344, 365, 386, 407, 428, 449, 470, 491, 512, 533, 554, 575, 596, 617, 638, 659, 931, 965, 999, 1033, 1067, 1101, 1135, 1169, 1203, 1237, 1271

Offset: 1

Charles R Greathouse IV, Nov 13 2014

nonn

It appears that a(n) exists for each n, and that the sequence is increasing.

			a(3) = 7 because the Fibonacci-like sequence 2, 1, 3, 4, 7, 11, ... contains 7 and the sum of the first two terms is 3, while no smaller sequences work. (All terms must be nonnegative.)

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000
Index to sequences related to the complexity of n

Cf. A249783, A054495.

A249783[n_] := A249783[n] = Module[{a, k, A, B}, If[n<2, Return[n]]; For[k = 1, k <= n-1, k++, For[a=0, a <= k-1, a++, A = a; B = k-A; While[BA249783[k] == n, Return[k]]]; Array[a, 50] (* Jean-François Alcover, Jan 06 2017, adapted from PARI *)

A054495(n)=fordiv(n,d,if(A010056(n/d),return(d)))
A249783(n)=if(n<2,return(n));for(k=1,n-1,for(a=0,k-1,my(A=a,B=k-A);while(B=least&&t<=#v&&v[t]==0,v[t]=n;while(v[least],if(least++>#v,return(v)))))

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

Original entry on oeis.org

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 *)

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

Original entry on oeis.org

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 *)

A341456 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) + t(1) + t(2) for an element t of T containing n.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Feb 12 2021

nonn

This sequence is a variant of A249783; here we consider tribonacci-like sequences, there Fibonacci like sequences. The scatterplots of these sequences both present polygonal shapes emerging from the origin.

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

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Scatterplot of the first 10000000 terms
Rémy Sigrist, PARI program for A341456

Cf. A000073, A001590, A213816, A249783.

PARI
```
See Links section.
```

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

A341285 Let B be the set of sequences of positive integers {b(k), k >= 0} such that for some k > 0 (necessarily unique) and any m >= 0, b(m+k) = b(m) + b(m+1) + ... + b(m+k-1); let g(b) = b(0); a(n) is the least value of g(b) for an element b of B containing n.

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist, Feb 16 2021

nonn

This sequence is a generalization of A249783 and A341456 to the set of "k-bonacci sequences of positive integers".

			The first terms of the elements b of B such that g(b) <= 3 are:
  g(b)  b(0)  b(1)  b(2)  b(3)  b(4)  b(5)  b(6)  b(7)  b(8)  b(9)
  ----  ----  ----  ----  ----  ----  ----  ----  ----  ----  ----
     1     1     1     1     1     1     1     1     1     1     1
     2     2     2     2     2     2     2     2     2     2     2
     2     1     1     2     3     5     8    13    21    34    55
     3     3     3     3     3     3     3     3     3     3     3
     3     1     2     3     5     8    13    21    34    55    89
     3     2     1     3     4     7    11    18    29    47    76
     3     1     1     1     3     5     9    17    31    57   105
- so a(1) = 1,
     a(2) = a(3) = a(5) = a(8) = 2,
     a(4) = a(7) = a(9) = a(11) = a(17) = a(18) = 3.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C program for A341285

Cf. A020695, A070939, A249783, A341456, A341699.

C
```
See Links section.
```

a(n) <= n.
a(m*n) <= m*a(n).
a(n) = 2 iff n belongs to A020695.
a(n) = A070939(A341699(n)).

cp's OEIS Frontend

A246399 Least k such that A249783(k) = n.

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

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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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A341456 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) + t(1) + t(2) for an element t of T containing n.

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A341285 Let B be the set of sequences of positive integers {b(k), k >= 0} such that for some k > 0 (necessarily unique) and any m >= 0, b(m+k) = b(m) + b(m+1) + ... + b(m+k-1); let g(b) = b(0); a(n) is the least value of g(b) for an element b of B containing n.

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