Chain fraction [0; 𝑎1, 1, 𝑎3, ..., 𝑎𝑛] and its stepwise approximation

Let 𝑂𝑋𝑌 be a rectangular coordinate system with an integer lattice, 𝐴𝐵 is a segment with integer ends 𝐴 = (𝑞; 0)𝑂𝑋𝑌 = 𝐴(𝑞) and 𝐵 = (0; 𝑝)𝑂𝑋𝑌 = 𝐵(𝑝) such that the decomposition of 𝑝
𝑞 into a continued fraction has the form [0; 𝑎1, 1, 𝑎3, . . . , 𝑎2𝑁′+1], where 𝑎− 1 , 𝑎3, . . . , 𝑎2𝑁′+1, 𝑁′, 𝑝, 𝑞,𝑁′ ∈ N : (= {1, 2, . . .}), moreover, 𝑝 and 𝑞 are mutually simple.
Consider the set S𝐴𝐵 of such unit squares (cells) of this lattice, that the interior of each of them has a nonempty intersection with 𝐴𝐵. The boundary of this set is represented as a union
of polylines S− 𝐴𝐵 and S+ 𝐴𝐵 such that their extreme vertices are the points 𝐴 and 𝐵. Here, the minus index (plus) indicates that S− 𝐴𝐵 (S+ 𝐴𝐵) lies on the left (right) side of the segment 𝐴𝐵
when moving from 𝐴 to 𝐵.
Polyline S− 𝐴𝐵 (S+ 𝐴𝐵)let’s call left (right) (integer) stepwise approximation of the segment 𝐴𝐵 or continued fraction [0; 𝑎1, 1, 𝑎3, . . . , 𝑎𝑁]. For brevity, we will call such polylines marches, and
their horizontal links—steps, vertical ones—heights, which is caused by association with a flight of stairs. The following results were obtained in the work:
1. Formulas for integer approximation of the stucco fraction, both finite and infinite.
2. Algorithmic construction of marches S± 𝐴𝐵, determined by the considered chain fraction. gives their analytical task through the marches of vectors-terms of the second equality:

where the vectors 𝑒2𝑁′−1 = (−𝑞2𝑁′−4; 𝑝2𝑁′−4)𝑂𝑋𝑌 and 𝑒2𝑁′ = (−𝑞2𝑁′−3; 𝑝2𝑁′−3)𝑂𝑋𝑌 , whose coordinates are determined by suitable fractions 𝑝𝑛−3/𝑞𝑛−3 of the order 𝑛 = 2𝑁′− − 1.2𝑁′, the original continued fraction, and the points of application of the vectors are indicated in parentheses.
3. The location of the steps of width 𝑎1 in S− 𝐴𝐵, is set using a family of parallel transfers of a segment with ends (𝑞 − 𝑎1; 0)𝑂𝑋𝑌 and (𝑞 − 1; 0)𝑂𝑋𝑌 . We find each member of the family through approximation formulas for [0; 𝑎1, 1, 𝑎3, . . . , 𝑎𝑁].

4. The formula giving the number of steps of width 𝑎1 in S−
𝐴𝐵 is a tridiagonal determinant (2𝑁′ − 5)-th order: the main diagonal consists of elements 𝑎4, 1, 𝑎5, . . . , 𝑎2𝑁′−6,the upper
diagonal is — of ones, the lower one is minus ones, and the rest are zeros.

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