## Shortcut to Numeric Values of the Coefficients in the *dn+c*
Formulas

From the experience with the left descent to
27 and its indication that 2^*a**3^*b***n*+*c*
is the form of the descriptors of the several members of the left
descent, and the experience with a number of simpler left descents,
*via*
downward development and upward
development, we can establish some general rules for the formation
of the 2^*a**3^*b* coefficients.

We call the coefficients *x(i)* where *i* runs from 1 for the
extension element which heads the left descent to *omega* for the
leaf element which closes the left descent. The coefficients all have
both a non-zero power of 2 and a non-zero power of 3 as factors. Each
rule below completely determines the power of the 2 or 3 it is concerned
with -- there is no effect on the value from one rule by another rule.
The resulting coefficients are those of left descents anchored at both
ends.

(1) The *x(1)* coefficient has 2^3 as a factor.

(2) The *x(1)* coefficient has 3^*omega* as a factor.

(3) The exponent in the 2^*a* factor in the coefficients goes up
by 2 for each *b* step and by 1 for each *s* step from
*x(1)* to *x(omega)* as the descent is traversed downwards.

(4) The exponent in the 3^*b* factor in the coefficients goes
down 1 in each step from *x(1)* to *x(omega)* as the descent is
traversed downwards. (See the end of this page for a corollary.)

(5) The constant *c* is correctly calculated by Maple and/or by
a process like that illustrated above.

Applying these rules in the *esbt* case, where *omega* is 3,
we get:

coefficient of *x(1)* will be 2^3*3^3 or 216,

coefficient of *x(2)* will be 2^4*3^2 or 144, and

coefficient of *x(3)* will be 2^6*3^1 or 192,

in complete
accord with the more arduously derived coefficients by the downward development and upward
development.

These rules were employed in the treegrow
program to avoid massive amounts of multiple precision calculation.
The program produced >280,000 lines of output, each giving the
formula representing a particular element in a particular left descent
assemblage. All the *c*s were unique in the file. The file was sorted
to produce a table of cardinalities of formulas with identical
coefficients. I would be happy to share this file with anyone interested. It is a
15M file but pkarcs to about 3.4M.
There's a corollary, belatedly discovered from observation of Figure 9, the example illustrating the location of an
arbitrary odd integer in the abstract predecessor tree. Comparison of the
coefficients in columns 6 and 7 show that each generation sees an increase
in their ratio by a factor of 3 from the bottom up. I.e., each step of
the bottom-up subsetting process identifies subsets which are 1/3^n of the
size of the entire node contents counting from the bottom up. This merely
reflects the division into thirds which each step in the abstract
predecessor tree development incurs.

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