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COMMENT
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The frequency of each distinct term (1,2,3 or 4) tends
to converge to the ratio of each diagonal, (a,b,c, or d) to the sum of
the 4 diagonal lengths. The four nonagon diagonals are a = 1, b = 1.87938524...,
c = 2.5320888862... and 2.87938524157, with the sum a+b+c+d = d^2 = 8.2985936...Converting
these terms to percentages, a = .120614..., b = .226681..., c = .305407...,
and d = .347296. Through n = 45, we can thus expect 16 4's (correct), since
round:(45*.347296...) = 16. The numbers of terms in each subset strung together
is found in A006357: (1, 4, 10, 30, 85...), thus: (4); (1,2,3,4); (4,3,4,2,3,4,1,2,3,4);...;
while the distributive breakdown of numbers of 1's, 2's, 3's and 4's may
be found in the 4-termed set of vectors in A076264: 1 1 1 1 4 3 2 1 10 9
7 4 30 26 19 10 ..where the sum of 4 terms in a row = the left term in the
next row. For example, the frequency distribution of 30 includes ten 4's,
nine 3's, 7 2's and four 1's. Check: the subset of 30 terms generated from
the previous subset of 10: (1,2,3,4,2,3,4,1,2,3,4,3,4,2,3,4,1,2,3,4,4,3,4,2,3,4,1,2,3,4).
A fractal structure is suggested by parsing each subset into groups: ((1,2,3,4),
(2,3,4), (1,2,3,4), (3,4), (2,3,4), (1,2,3,4), (4), (3,4), (2,3,4), (1,2,3,4).
That is, 10 groups: four with four terms, three with three terms, two with
two terms and one with one term. Replacing the terms (4,3,2,1) with the
diagonal lengths (d,c,b,a) and referring to the set of vectors: (1,1,1,1;
4,3,2,1; 10,9,7,4;...), label these rows 2,3,4...and consider (2,3,4...)
exponents to diagonal d: 2.87938524...; such that for example, "4" corresponds
to (10,9,7,4); and (Cf. Steinbach), d^4 = 68.738349...= (10*d + 9*c + 7*b
+ 4*a). Such relationships are a consequence of the "Diagonal Product Formulas" mentioned on p. 23.
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