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 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 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:=n:U:=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:=U:                     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

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Last modified September 21 19:57 EDT 2020. Contains 337273 sequences. (Running on oeis4.)