We'll present a series of tables and a figure to explain in greater detail how the integer densities in the collection of subsets obtained by detailed dissection of the l.d.a.s in the abstract tree lead to the conclusion that all odd positive integers appear in it.

First, we had better be sure we understand how the elements of
the l.d.a.s which share a *d*(=2^*a**3^*b*) value map
into the cardinality table which provides the basis for summation of the
appropriate densities to determine the total density of odd integers in
the tree. The following table expands the upper left 3 by 3 corner
of the cardinality table by indicating the l.d.a. elements
contributing to the cardinality. The l.d.a.s appear in
e(s|b)_{k}t notation and the depth of each element in its
l.d.a. appears in parentheses.

Note that the et , est, ebt, and esst l.d.a.s are
the only ones whose entire contents are presented, because the
later elements of the other l.d.a.s mentioned contain
*a*>6, outside of the pictured fragment. The
bracketed *d* values define the density of integers as
1/*d* (among all integers) or 2/*d* (among
only the odd integers). As is basic to the very existence of the
cardinality table, the number of entries in each box determines
the entry in that position in the cardinality table. Thus,
the entries in each box constitute the complete set of odd integers
contained by the set of l.d.a. elements sharing a *d* value in the
finite interval denoted in brackets.

The density contributed within the intervals in this first table
establish the densities these l.d.a. elements will provide within
larger intervals provided that the larger intervals are integer
multiples of the initial ones. Fortunately, the widths
occupied by the sets of sets as *a* is increased across the rows of
the cardinality table do always arise as multiples by 2 in the row
dimension or, when *b* is increased in the column
dimension, as multiples by 3. We will synchronize the
intervals covered by each subset by starting them all at 3, thus
assuring that all of them appearing in any given row or column will come
to a common terminus at the end of the largest of them. To
underscore this point, the densities are expressed in unreduced
fractions in Tables 2, 3, and 4. The use of
unreduced fractions reflects the patterns initiated by,
and
evident in, this first table and demonstrates the set width growth by
multiples of 2 or 3 as required to justify the summation of
densities in finite sets.

## Table 1. L.d.a. Element Contents of Cells for 3<=a<=5 and 1<=b<=4

from a=3 from a=4 from a=5 ---------|------------------|---------------------|------------------------------------------ from b=1 |et(1) 21[24] |est(2) 3[48] |ebt(2) 81[96] esst(3) 87[96] ---------|------------------|---------------------|------------------------------------------ from b=2 |est(1) 5[72] |esbt(2) 115[144] |essst(3) 23[288] essbt(3) 151[288] |ebt(1) 61[72] |esst(2) 131[144] |ebst(2) 113[288] ebbt(2) 241[288] ---------|------------------|---------------------|------------------------------------------ from b=3 |ebst(1) 85[216] |esbbt(2) 19[432] |esssbt(3) 119[864] ebsbt(2) 497[864] |esbt(1) 173[216] |essst(2) 35[432] |essbst(3) 247[864] ebbst(2) 625[864] |ebbt(1) 181[216] |essbt(2) 227[432] |ebsst(2) 305[864] ebbbt(2) 721[864] |esst(1) 197[216] |esbst(2) 355[432] |essbbt(3) 343[864] esssst(3) 791[864] ---------|------------------|---------------------|------------------------------------------ from b=4 |esbbt(1) 29[648] |esbsst(2) 67[1296] |ebssbt(2) 17[2592] essbbst(3) 631[2592] |essst(1) 53[648] |esssbt(2) 179[1296] |ebbsbt(2) 49[2592] esssbst(3) 695[2592] |ebsst(1) 229[648] |esbsbt(2) 211[1296] |essbbbt(3) 55[2592] ebbsst(2) 1201[2592] |essbt(1) 341[648] |esbbst(2) 307[1296] |ebbbst(2) 145[2592] ebsbbt(2) 1649[2592] |ebsbt(1) 373[648] |essbst(2) 371[1296] |ebsbst(2) 209[2592] essbsst(3) 1687[2592] |ebbst(1) 469[648] |essbbt(2) 515[1296] |essssst(3) 215[2592] esssbbt(3) 2135[2592] |esbst(1) 533[648] |esbbbt(2) 1027[1296] |essssbt(3) 503[2592] ebbbbt(2) 2161[2592] |ebbbt(1) 541[648] |esssst(2) 1187[1296] |essbsbt(3) 535[2592] ebssst(2) 2321[2592]

We have elsewhere presented tables illustrating the details of the downward and upward stages in the dissection of the elements in each branch of the abstract tree into disjoint subsets which each represent the contents of some one single element of some one l.d.a. But those tables do not make it immediately obvious how the densities of the dissected subset are reduced in both the downward and the subsequent upward processes. Accordingly, an additional diagram is now presented in which the width of the subsets represent their densities. Although the scaling of the top row differs from that of the other two rows and the depth reached in its development of the abstract tree is very limited, the diagram makes it clear that the subsetting occurs during both directions of travel, resulting in the very fine dissection which is shown in the above table.

It remains to show that the densities represented by all the sets of
elements of all the l.d.a.s indicate that all the odd integers are
included in the abstract tree. To this end, three additional tables
will be presented. Note that the densities employed are
1/*d* (appropriate to consideration of odd positive integers only)
throughout. We will be able to present data describing a
somewhat more liberal selection from the space of the cardinality table
-- a 7 by 6 (or 5 by 5) corner from the upper left.

Table 2 shows the density among the odd
integers of integers provided by every (*a,b*)
pair. Remember that the densities cited are those defined within
the finite (2^*a**3^*b*) width of the first
instance of the contained sets of l.d.a.s elements, but
that, conveniently enough, later sets of elements do occur
with a width obtained by multiplication by 2 or 3. The densities
are presented as fractions which have not been reduced to lowest
terms, because their pattern is more easily discerned in that
way. These densities are calculated from the formula involving the
Fibonacci series (F(i)).

Using 2^{b-a}*F(a-2)/3^{b}, the following table
results.

## Table 2. Density of Odd Integers for Cells 3<=a<=9 and 1<=b<=6

b\a 3 4 5 6 7 8 9 1 1/12 1/24 2/48 3/96 5/192 8/384 13/768 2 1/18 1/36 2/72 3/144 5/288 8/576 13/1152 3 1/27 1/54 2/108 3/216 5/432 8/864 13/1728 4 2/81 2/162 4/324 6/648 10/1296 16/2592 26/5184 5 4/243 4/486 8/972 12/1944 20/3888 32/7776 52/15552 6 8/729 8/1458 16/2916 24/5832 40/11664 64/23328 104/46584

Table 3 shows the row-cumulative values of the densities across a
constant 3^*b* value. Again, fractions which have not
been reduced to their lowest terms are employed. The cumulative
densities from *a* to *at* in row *b* are given,
again employing the Fiboniacci series, by using

add(2the following table results.^{b-a}*F(a-2)/3^{b}),a=a..at #row cumulative

## Table 3. Cumulative Row Sums of Density for Cells 3<=a<=9 and 1<=b<=6

b\a 3 4 5 6 7 8 9 asymptote 1 1/12 1/8 1/6 19/96 43/192 47/192 67/256 ... 1/3 2 1/18 1/12 1/9 19/144 43/288 47/288 67/384 ... 2/9 3 1/27 1/18 2/27 19/216 43/432 47/432 67/576 ... 4/27 4 2/81 3/81 4/81 19/324 43/648 47/648 67/864 ... 8/81 5 4/243 8/243 8/243 19/486 43/972 47/972 67/1296 ...16/243 6 8/729 12/729 16/729 19/729 43/1458 47/1458 67/1944 ...32/729

The values calculated by formula for *at*=100 and *at*=200 had approached
the asymptotic value to about 56 parts in 10^11 and 23 parts in 10^20,
respectively. Finally, from these asymptotic values, the
sum((2^(*i*-1)/3^*i*),*i*=1..infinity) is 1, showing that all the odd integers
are included in the asymptote based on increasing widths of finite sets.

Finally, the Table 4 (showing a 5 by 5 chunk of the territory) shows a column-cumulative value for each cell from the cardinality table. Using

add((2the following table results.^{b-a}*F(a-2)/3^{b}),b=b..bt) #column cumulative

## Table 4. Cumulative Columns Sums of Density for Cells 3<=a<=7 and 1<=b<=5

b\a 3 4 5 6 7 1 1/12 1/24 2/48 3/92 5/192 2 5/36 5/72 10/144 15/288 25/288 3 19/108 19/216 38/532 57/864 95/1728 4 65/324 65/648 130/1296 195/2592 325/5984 5 211/972 211/1944 422/3848 633/7776 1055/15552 asymptote: 1/4 1/8 2/16 3/32 5/64

The values calculated by formula for *bt*=100 and *bt*=200 had approached
the asymptotic value to about 13 parts in 10^19 and 16 parts in 10^37,
respectively. These asymptotic values constitute the expression
sum((F(*i*)/2^(*i*+1),*i*=1..infinity) whose value is known to be 1, in
agreement with the result from the row sums, and, once again, indicating
that in the asymptote all odd positive integers appear in the abstract
tree.

If you prefer MapleV4's opinion to my "engineering" approach, consider the following fragment from a MapleV4 session. The same expressions used above to gather the row-cumulative and the column-cumulative finite sums of the densities in the cardinality table are used again, but this time to determine the limit to the contribution of the densities as the initial row/column is increased to infinity. Once again, it appears that in the limit all odd positive integers appear in the abstract predecessor tree.

> f := proc(n) option remember; > if n<2 then n else f(n-1)+f(n-2) fi end; f := proc(n) option remember;if n < 2 then n else f(n - 1) + f(n - 2) fi end > a:=3; b:= 1; a := 3 b := 1 > limit(2^(bt-a)*f(a-2)/3^bt,bt=infinity,left); #row cumulative 0 > limit(2^(b-at)*f(a-2)/3^b,at=infinity,left); #column cumulative 0