A271082 Triangle read by rows, the coefficients of the (3x+1)-polynomials.

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

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

Links

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

Michel Lagneau, Coefficients

Example

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                              |
+----+-----------------------------------------------------------+

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