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 cs 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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