login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A271082 Triangle read by rows, the coefficients of the (3x+1)-polynomials. 1
1, -3, 3, 1, -30, 5, -15, 7, 1, 2, 4, 16, -1920, 9, 1, 4, 8, 16, 64, -7680, 11, 1, 2, 8, -960, 13, 1, -120, 15, 1, 2, 4, 8, -3840, 17, 1, 4, -480, 19, 1, 2, 16, 32, 128, -15360, 21, -63, 23, 1, 2, 4, -1920, 25, 1, 4, 8, 64, 128, 512, -61440 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Definition of the (3x+1)-polynomials.

The 3x+1 problem is an exceptional case of the zx + 1 problem (for z real or complex). We associate each odd integer x with a polynomial f(z) whose roots have the same behavior as the integer 3 in the 3x + 1 problem.

The polynomial f(z) is called "(3x+1)-polynomials" and the problem zx + 1 generates the same number of iterations as the 3x + 1 problem requires to reach 1. The polynomial f(z) has interesting properties, for instance the study of the roots of f(z)= 0.

The following example shows the process.

Let’s consider x = 17. The corresponding reduced Collatz trajectory containing only odd numbers (17, 13, 5, 1) is obtained from the following steps:

start with x = 17;

step 1:  (3*17 + 1)/4 = 52/4 = 13;

step 2:  (3*(3*17 + 1)/4 + 1)/8 = 40/8 = 5;

step 3:  (3*(3*(3*17 + 1)/4 + 1)/8 + 1)/16 = 16/16 = 1.

Step 4:  substitute the number 3 by the variable z. So, we obtain the following equation:

f(z) = 17z^3 + z^2 + 4z - 480 = (z-3) g(z) = (z-3)(17z^2 + 52z + 160)= 0.

We would consider that the polynomial f(z) is associated with the integer 17.

The three roots are:

z0 = 3;

z1 = -1.529411765 + 2.659448131 I;

z2 = -1.529411765 - 2.659448131 I.

The roots z1 and z2 have the same behavior as the integer z0=3, and the 3*x + 1 problem, z1*x + 1 problem and z2*x + 1 problem are identical for x = 17 : we obtain the same number of iterations of the reduced Collatz function required to yield 1: 17 = 2*9-1 => A075680(9) = 3 iterations.

For example, with z1 we obtain the following steps:

(17*z1 + 1)/4 = -6.250000001 + 11.30265455*I

(z1*(17*z1 + 1)/4 + 1)/8 = -2.437500001 - 4.238495460*I

(z1*(z1*(17*z1 + 1)/4 + 1)/8 + 1)/16 = 1.

For each number x = 2n-1, if the Collatz conjecture is true, the polynomial f(z) is of the general form :

f(z) =(2n-1)*z^p + z^(p-1) + 2^a*z^(n-2) + 2^b*z^(n-3) + ... + 2^w*z + 2^r - 2^s = (z-3) g(z) with the property : degree(f(z)) = p = A075680(n), n>1.

s is the number of divisions by 2 at the last step

r is the number of divisions by 2 at before the last step

a is the number of divisions by 2 at the first step

b is the number of divisions by 2 at the second step

.............................................

Triangle begins:

1, -3,

3, 1, -30,

5, -15,

7, 1, 2, 4, 16, -1920,

9, 1, 4, 8, 16, 64, -7680,

11, 1, 2, 8, -960,

13, 1, -120,

15, 1, 2, 4, 8, -3840,

17, 1, 4, -480,

19, 1, 2, 16, 32, 128, -15360,

21, -63,

23, 1, 2, 4, -1920,

25, 1, 4, 8, 64, 128, 512, -61440,

The corresponding polynomials are:

+----+-----------------------------------------------------------+

|  x | Polynomials f(z) including the factor (z - 3)             |

+----+-----------------------------------------------------------+

|  1 |  z - 3                                                    |

|  3 |  3z^2 + z - 30                                            |

|  5 |  5z - 15                                                  |

|  7 |  7z^5 + z^4 + 2z^3 + 4z^2 + 16^z - 1920                   |

|  9 |  9z^6 + z^5 + 4z^4 + 8z^3 + 16z^2 + 64z - 7680            |

| 11 |  11z^4 + z^3 + 2z^2 + 8z - 960                            |

| 13 |  13z^2 + z -120                                           |

| 15 |  15z^5 + z^4 + 2z^3 + 4z^2 + 8z  - 3840                   |

| 17 |  17z^3 + z^2 + 4z - 480                                   |

| 19 |  19z^6 + z^5 + 2z^4 + 16z^3 + 32z^2 + 128z - 15360        |

| 21 |  21z - 63                                                 |

| 23 |  23z^4 + z^3 + 2z^2 + 4z - 1920                           |

.................................................

+----+-----------------------------------------------------------+

|  x |            Polynomials f(z)/(z - 3)                       |

+----+-----------------------------------------------------------+

|  1 |  1                                                        |

|  3 |  3z + 10                                                  |

|  5 |  5                                                        |

|  7 |  7z^4 + 22z^3 + 68z^2 + 208z +640                         |

|  9 |  9z^5 + 28z^4 + 88z^3 + 272z^2 + 832z + 2560              |

| 11 |  11z^3 + 34z^2 + 104z + 320                               |

| 13 |  13z + 40                                                 |

| 15 |  15z^4 + 46z^3 + 140z^2 + 424z + 1280                     |

| 17 |  17z^2 + 52z + 160                                        |

| 19 |  19z^5 + 58z^4 + 176z^3 + 544z^2 + 1664z + 5120           |

| 21 |  21                                                       |

| 23 |  23z^3 + 70 z^2 + 212z + 640                              |

...............................................

LINKS

Table of n, a(n) for n=1..60.

Michel Lagneau, Coefficients

MAPLE

for m from 1 by 2 to 27 do:    T:=array(1..50, [0$50]):U:=array(1..50, [0$50]):

n:=m:ii:=2:xx1:=2:pp1:=0:s:=0:U[1]:=n:U[2]:=1:

     for q from 1 to 100  while(xx1<>1)do:

       n1:=3*n+1:

        for p from 1 to 50 do:

         p1:=2^p:x1:=floor(n1/p1):x0:=irem(n1, p1):

          if x0=0 and xx1<> 1

           then

           pp1:=p:xx1:=x1:

           else

          fi:

        od:

         T[ii]:=pp1:n1:=x1:n:=xx1:ii:=ii+1:od:s:=0:

                 for j from 1 to ii-3 do:

                   s:=s+T[j]:U[j+2]:=2^s:

                 od:

                   s:=s+T[ii-2]:s1:=2^s:s:=s+T[ii-1]:

                   s2:=2^s:U[ii]:=s1-s2:

                   W:=array(1..ii-1, [0$ii-1]):

                   W[1]:=U[1]:

                    for l from 2 to ii-1 do:

                     W[l]:=U[l+1]:

                    od:

                    print(m):

                    print(W):

   od:

CROSSREFS

Cf. A075680, A171870.

Sequence in context: A318110 A117262 A065431 * A053375 A117252 A332498

Adjacent sequences:  A271079 A271080 A271081 * A271083 A271084 A271085

KEYWORD

sign,tabf

AUTHOR

Michel Lagneau, Mar 30 2016

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 2 18:09 EDT 2020. Contains 334787 sequences. (Running on oeis4.)