By rewriting the relation as , a right triangle is looked at. Some geometrical observations in connection with plane parqueting lead to an inductive sequence of right triangles with as initial one converging to the segment of the real line. The sequence of their hypotenuses forms a sequence of real numbers which initiates some beautiful algebraic patterns. They are determined through some recurrence relations which are proper for being evaluated with computer algebra.
1 An inductive sequence of positive numbers
With the initial numbers and a real sequence is defined inductively via
By introducing positive numbers by , , a sequence of circles in the complex plane is given which is part of some parqueting of the complex plane . The parqueting-reflection principle is always initiating iteratively given sequences of complex numbers; see  for another sample. The sequence presented here served in [1, 3] for treating problems in a hyperbolic domain. The triangles appear by taking one tangent to the unit circle through the center of a circle, the radius of the unit circle perpendicular to the tangent, and the segment as the hypotenuse; see Figure 1.
The sequence is monotonic decreasing with limit 1. In particular,
and in particular
follow. The last equation shows , and hence the do not depend on k:
The inequalities imply
for any .
The monotonicity is seen from
together with . ∎
The sequence allows a simpler representation.
For any ,
By simple verification,
is seen. Combining this with
in the definitions for and shows
Thus is independent of k, and hence coincides with the value for . ∎
Obviously, . Lemma 1.2 suggests a new definition of the sequence with just one initial value and
The first members of the sequences and are presented in Section A.
2 A recurrence relation
With the sequence and its related coefficients , the system of recurrence relations
defines two new sequences with the initial values , , , . The first further members of the sequences are listed in Section A.
By using the relation according to Lemma 1.2, these sequences are given in a shorter form as
In order to show that these formulas present solutions to the recurrence relations, equation (2.1) for is proved, assuming the expressions for and are verified already. Then
Splitting the -term from the first sum and multiplying shows
The first three terms on the right-hand side form
The next two sums are composed to
By leaving the next sum unchanged, the last two become
In the same manner, the other three formulas can be verified. ∎
The expressions for and in Theorem 2.1 show that these quantities are combinations of products of the . That they in fact form very regular and beautiful algebraic combinations can be seen by writing these expressions down explicitly. These formulas of arbitrary order can even be produced by computer manipulations on the basis of the recurrence relations (2.1) and (2.2). Moreover, the and can in the same way be expressed through the and . Also, these formulas show a very regular and beautiful algebraic structure; see A.3 and A.4.
As the sequences , , also , converge to 0, but only the first two are monotone decreasing.
The sequences and are monotone. They and the sequences and converge to 0.
From the monotonicity of , the estimations
are obvious. That and are null-sequences follows from . The other two sequences consist of non-negative numbers. This is seen from (2.1) and (2.2) as the initial values are non-negative and the are positive numbers. From the assumption
by using (1.1) reformulated as
both right-hand sides from
can be estimated from above by
is less than 1 for , as can be seen from
The assumptions made are readily satisfied for . ∎
3 A second recurrence relation
For , the terms
satisfy the systems
A direct consequence from Theorem 2.1 is the next statement.
Exemplarily, the second formula will be verified from
separating the factor and renaming summation indices lead to
In a similar way, is split:
Adding the two formulas gives the expression for . ∎
As the , , also , converge to 0.
Remark 2.2 also applies for the and . They are expressible either through the or the and ; see A.5 and A.6. But neither the formulas in Theorem 2.1 nor those in Theorem 3.1 reveal the beauty and regularity of the algebraic pattern. But writing these formulas out or simpler using some computer algebra to develop the single representations on the basis of the respective recurrence relations (2.1) and (2.2) or (3.1) and (3.2) unveils their symmetry.
By the relations
is eliminated as parameter. From (3.2), then
respectively. Shifting the summation for the second term on the right-hand side replacing λ by gives
In the same way, the part for can be handled. By repeating the procedure, the respective formulas for the index can be achieved, completing the proof. ∎
Here the indices involved are partitions of the set
where the indices again are decompositions of , each subset ordered according to size.
creates similar but different expressions. By neglecting the right-hand sides from the last two equations, can be kept as parameter. By using the latter relations for starting, from formulas (3.1),
Reflecting the summation index, i.e. interchanging λ with , in the second part of gives for this part the same expression as in the first part, but the summation is taken between and . The indices now vary according to
In order to give an impression of the latter pattern, the first elements of the sequences , , , are also listed in Section A.
A Visualization of the sequences
expressed through :
expressed through :
expressed through :
expressed through :
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