I have already discussed the simplest
features of the 3^{j} systems, *j*>0. Since
interesting complications do arise,. but are complicated to demonstrate
and/or explain, a separate page about them is presented herewith.

Another instance of the predecessor tree, this time presented as a general
tree, arising for j=12, is presented. This example
further illustrates the great width and relatively shallow and relatively constant depth
of predecessor trees in this family. More particularly, close
examination reveals that the number of steps to leaves varies among the
the various paths from the root 531441 (3^{12}) to leaf nodes
(<>0[3]).

While it is true that there must be at least *j* steps to a leaf
in a tree formed for 3^{j}, it turns out that there
are frequently paths with length greater than *j*. For
example, in the tree rooted in 9, the path 9,
189, 123, 79 fails to consume a power of three at its first
step so the path has length 3 instead of 2. In the same
tree, the path 9, 45, 27, 15, 7 fails to
consume a power of 3 in its first 2 steps, resulting in a path
length of 4 instead of 2.

There is a spectacular example of a lengthy sequence of steps which
keep a constant power of 3 as the steps progress. The example was
discovered at 103 during the scan of the integers from 1 to 201 for
(3*n*+3^{2})/2^{i} for convergence to the root. As
mentioned
elsewhere, this simple scan served to make a preliminary
separation between Collatz-like and Collatz-unlike behavior, and no
untoward behavior was found.

The spectacular path in the (3n+9)/2^{i} predecessor
tree has a leaf node (103) and its two successors (159, 243) at the
bottom and reaches 45 and 9 at the root, with 40 steps between which leave
the power of 3 in the c part of the *dn*+*c* set
representation untouched. I have factored 9 out of all the *c* values
in the following table to make the numbers easier to see and to make it plain
that the same power of 3 remains in the *c* values for this whole part of
the trajectory. The 5*9 at the top represents 45 (just short of the
root) and the 27*9 at the bottom represents 243, two steps
(each of which does consume a power of 3) up from the leaf node for this
path.

From this discovery as a starting point, more detail was obtained from
MapleV4 by having it solve the set of simultaneous equations shown in the left
column followed by extensive reordering, reformatting, and
additional processing. MapleV4 gave the results in the central
column, from which the large coefficients were extracted and
factored, again using MapleV4. The factors of the large coefficients
are shown in the last column. The decrement in the power of 3 in *d*
at every stage combined with the increment in the power of 2 which agrees with
the value of *i* in the 2^{i} part of the defining iteration is
familiar from the experience with
(3*n*+1)/2^{i}. It
is evident that this path recurs at extremely large intervals.

Remarkably, this itinerary, with the factor 9 removed from the *c* term
starting at 243, the third element of the successor sequence starting from 103,
is precisely the itinerary from 27 to 1 in the (3n+1)/2^{i} case.

isolve(setx3p2); 40 setx3p2:={ { n0= 5*9+ 12157665459056928801_N1, (3) 5 39 n1=(32*n0-1)/3, n1= 53*9+ 129681764896607240544_N1, (2) (3) 6 38 n2=(2*n1-1)/3, n2= 35*9+ 86454509931071493696_N1, (2) (3) 7 37 n3=(2*n2-1)/3, n3= 23*9+ 57636339954047662464_N1, (2) (3) 10 36 n4=(8*n3-1)/3, n4= 61*9+ 153696906544127099904_N1, (2) (3) 14 35 n5=(16*n4-1)/3, n5= 325*9+ 819716834902011199488_N1, (2) (3) 16 34 n6=(4*n5-1)/3, n6= 433*9+1092955779869348265984_N1, (2) (3) 18 33 n7=(4*n6-1)/3, n7= 577*9+1457274373159131021312_N1, (2) (3) 22 32 n8=(16*n7-1)/3, n8=3077*9+7772129990182032113664_N1, (2) (3) 23 31 n9=(2*n8-1)/3, n9=2051*9+5181419993454688075776_N1, (2) (3) 24 30 n10=(2*n9-1)/3, n10=1367*9+3454279995636458717184_N1, (2) (3) 25 29 n11=(2*n10-1)/3, n11= 911*9+2302853330424305811456_N1, (2) (3) 28 28 n12=(8*n11-1)/3, n12=2429*9+6140942214464815497216_N1, (2) (3) 29 27 n13=(2*n12-1)/3, n13=1619*9+4093961476309876998144_N1, (2) (3) 30 26 n14=(2*n13-1)/3, n14=1079*9+2729307650873251332096_N1, (2) (3) 31 25 n15=(2*n14-1)/3, n15= 719*9+1819538433915500888064_N1, (2) (3) 32 24 n16=(2*n15-1)/3, n16= 479*9+1213025622610333925376_N1, (2) (3) 33 23 n17=(2*n16-1)/3, n17= 319*9+ 808683748406889283584_N1, (2) (3) 35 22 n18=(4*n17-1)/3, n18= 425*9+1078244997875852378112_N1, (2) (3) 36 21 n19=(2*n18-1)/3, n19= 283*9+ 718829998583901585408_N1, (2) (3) 38 20 n20=(4*n19-1)/3, n20= 377*9+ 958439998111868780544_N1, (2) (3) 39 19 n21=(2*n20-1)/3, n21= 251*9+ 638959998741245853696_N1, (2) (3) 40 18 n22=(2*n21-1)/3, n22= 167*9+ 425973332494163902464_N1, (2) (3) 43 17 n23=(8*n22-1)/3, n23= 445*9+1135928886651103739904_N1, (2) (3) 45 16 n24=(4*n23-1)/3, n24= 593*9+1514571848868138319872_N1, (2) (3) 46 15 n25=(2*n24-1)/3, n25= 395*9+1009714565912092213248_N1, (2) (3) 47 14 n26=(2*n25-1)/3, n26= 263*9+ 673143043941394808832_N1, (2) (3) 48 13 n27=(2*n26-1)/3, n27= 175*9+ 448762029294263205888_N1, (2) (3) 50 12 n28=(4*n27-1)/3, n28= 233*9+ 598349372392350941184_N1, (2) (3) 51 11 n29=(2*n28-1)/3, n29= 155*9+ 398899581594900627456_N1, (2) (3) 52 10 n30=(2*n29-1)/3, n30= 103*9+ 265933054396600418304_N1, (2) (3) 54 9 n31=(4*n30-1)/3, n31= 137*9+ 354577405862133891072_N1, (2) (3) 55 8 n32=(2*n31-1)/3, n32= 91*9+ 236384937241422594048_N1, (2) (3) 57 7 n33=(4*n32-1)/3, n33= 121*9+ 315179916321896792064_N1, (2) (3) 59 6 n34=(4*n33-1)/3, n34= 161*9+ 420239888429195722752_N1, (2) (3) 60 5 n35=(2*n34-1)/3, n35= 107*9+ 280159925619463815168_N1, (2) (3) 61 4 n36=(2*n35-1)/3, n36= 71*9+ 186773283746309210112_N1 (2) (3) 62 3 n37=(2*n36-1)/3, n37= 47*9+ 124515522497539473408_N1, (2) (3) 63 2 n38=(2*n37-1)/3, n38= 31*9+ 83010348331692982272_N1, (2) (3) 65 n39=(4*n38-1)/3, n39= 41*9+ 110680464442257309696_N1, (2) (3) 66 n40=(2*n39-1)/3 n40= 27*9+ 73786976294838206464_N1, (2) }; }

This close parallel of the paths between (3*n*+1)/2^*i* and
(3*n*+9)/2^*i*, differing only in that the factor of 9 is found in
the value of *c* at each stage extends to (3*n*+3)/2^*i*,
(3*n*+27)/2^*i*, (3*n*+81)/2^*i*, and
(3*n*+243)/2^*i*. In the *dn*+*c* descriptor of the
sets of integers reached, in each case, the *c* value
contains the factor equal to 3^*j* of the
(3*n*+3^*j*)/2^*i*, while the *d* value
remains constant across *j* at every level of descent in the
predecessor path. The number of additional descent steps beyond
the 40th differs from one another across the value of *j* before a
leaf node is reached. See below for further details.

The remarkable reversal of roles between 0[3] nodes on the one hand
and 1[3] and 2[3] nodes on the other hand is evident here.
In (3*n*+1)/2^*i* the former are leaf nodes and the latter internal
nodes in the path, but in the *j*>0 instances the 0[3] nodes are
internal and the leaf nodes are either 1[3] or 2[3].

This being so, paths in (3*n*+3^*j*)/2^*i* itineraries closely parallel to
paths in the (3*n*+1)/2^*i* appear to occur for every integer in 0[3], i.e.
from every leaf of the predecessor tree for (3*n*+1)/2^*i*. I've
checked relatively few, of course, but it isn't hard to show
that this must be true in general.

One could build an inductive proof that a step which exists in the
predecessor tree for any integer an+b for the Collatz sequence based on
(3*n*+3^*j*)/2^*i* must have a parallel step in 3*n*+3^(*j*+1)/2^*i*.

(a) Applying the first to *an*+*b* results in (3*an*+*b*)+3^*j*)/2.

(b) Applying the second to 3*an*+3*b* results in (3(3*an*+*b*)+3^(*j*+1))/2,
which is the same as 3(3(*an*+*b*)+3^*i*)/2 after factoring out the 3's.

(c) Thus, generally, piling on a factor of three produces exactly
parallel paths (may I say homologous paths?) in the
itineraries where *j*>1 analogous to the original Collatz itineraries
where *j*=0.

(d) Since the l.d.a.s define important components of
paths in the *j*=1 itineraries, we may conclude that homologues of l.d.a.s
must appear in the predecessor trees for *j*>1.

The 0[3] nodes are internal in the (3*n*+3^*j*)/2^*i*,
for the *j*>0 cases, so each
must have a further few nodes below it in the predecessor tree to reach
a leaf node in the (3*n*+3^*j*)/2^*i* analog. Thus, the main
bodies (so to speak, composed entirely of 0[3] nodes) of the
series of 3^*j* predecessor trees are homologous, differing in the
values of *c* and *d* of the *dn*+*c* formulation by
successive multiples of 3 as *j* is
increased by 1 through the series of analogs. The question is
where and how do all the odd 1[3] and 2[3] nodes attach in these
homologous
predecessor trees. The answer for
*j*=1 is that they show up in the extensions (right descents)
from each node of a left descent and, for *j*>1, the leaf nodes which
terminate every left descent.

For *j*>1, additional integers appear as tails leading to leaf nodes
from that value which was the leaf node at *j*=0 (times 3^*j*) to
terminate each left descent. The elements in these tails are often (but
not always) in {0[3] AND 5[8]}, thus drawing on the reservoir of values
which in the *j*=0 case are the headers of null l.d.a.s. The following
table shows the series of tails below 27 and its multiples of 3 in the
family of predecessor trees.

j= 0 1 2 3 4 5 6 7 8 9 10 11 27 81 243 729 2189 6561 19683 59049 177147 531421 1594323 4782969 53 159 477 1431 4293 12879 38637 115911 347733 1043199 3129597 103 309 927 2781 8343 25029 75087 225261 675783 2027349 197 591 1773 5319 15957 47871 143613 430839 1292517 367 1101 3303 9909 29727 89181 267543 802529 653 1959 5877 17631 52893 158679 476037 1063 3189 9567 28701 86103 258309 1397 4191 12573 37719 113157 607 1821 5463 16389 -5347 -16041 -48123 5[8] 7[8] 5[8] 7[8] 5[8] 7[8] 5[8] 7[8] 5[8] 7[8] 5[8]

The relationship between entries in one column and those displaced
downward by one in the next column is *n**2-3^*j*.
E.g., 81*2-3 gives 159 and 103*2-9 gives 197. The bottom row
contains the values modulo 8 of the internal elements in each column in
the path to the leaf. Half of the columns seem to use numbers from
the set of headers of null l.d.a.s.

The entries in columns *j*>0 remain in 0[3] until the leaf is
reached, where values in 1[3] or 2[3] are found. For *j*>=9, negative
numbers are encountered before the expected leaf nodes are reached. I
don't know what to make of that.
.

My Collatz Home Page Index to Terms Used