@Kitonum 

I can understand the logic behind this command structure. Learned something new again :-) .

@Christian Wolinski 

...I (a beginner in Maple) didn’t understand this. Just like a long time ago, I had thought of a so-called ‘do-loop’, as I remember it from the days of Algol and Fortran. This was supposed to generate and display a list of results for a selected index range. I would appreciate further help.

@Aliocha 

......a simple recurrence relation should be possible. Recalling an old exercise and deciphering my handwriting, it might look like this:   I(n+1)=2*I(n)-I(n-1)
As far as I can tell at a quick glance, this can be derived from @Rouben Rostamian's result. However, programming it in Maple is still too difficult for me as a beginner.

edited:

I overlooked a term in the recurrence relation. Using @Rouben Rostamian's notation, the recurrence is:
f(n+1,x)=2*f(n,x)-f(n-1,x)+(2/n)*sin(n*x)

@Mariusz Iwaniuk 

...because countably many solutions to the equation become visible.

@janhardo 

...given Green's theorem, the structure of the integrand in the problem's surface integral looked so tempting to me - appearing as the difference of two rational terms where the degree of the denominator is two higher than that of the numerator :-(
Your analytical approach is impressive :-) . I've learned something new about Maple.

@sand15 @vv 

...identify(1.53978589107117). It took a lot of effort - and my entire monthly Maple-AI credit allowance - to obtain the "closed-form" result. However, the solution path provided by the AI ​​strikes me as fragmentary and difficult to follow. That explains my unsuccessful attempt using the "Green's theorem → parameterize the boundary → residue" approach; after all, the structure of the problem seemed to suggest it. But thanks to your help, everything is sorted out now.

@nm 

......had Maple not provided a solution. Formally, the existence and uniqueness theorem again provides a highly simplified - and therefore brief - justification. For instance, in Pontryagin’s book "Ordinary Differential Equations" (Theorem 1, p. 3), a prerequisite is stated - as is the case in Kamke, etc. - that, in the real domain, the points (x, y) of the explicit right-hand side (and thus the points of the sought-after solution) must lie within a domain (an open set). However, the abscissa "infinity" lies on the real boundary and is not part of the domain. Maple seems to be aware of this ;-) . In my experience, initial conditions of the unusual type you have presented always require special handling on a case-by-case basis. And dealing with this ODE in the complex plane turns into quite an adventure...

p. s.

Equations (ODEs, PDEs, integral equations)—alongside minimization problems in the calculus of variations—are commonly used to model real-world states or processes. In any such case, however, it is questionable whether "infinity" in initial or boundary conditions accurately models vast spatial extents or long time intervals. In the aforementioned problem, the solution along the ordinate fails to satisfy the initial condition after the limit a-->oo is taken. This serves as an example of how convergence from a finite domain toward the boundary of a Lipschitz domain does not yield a solution to the initial value problem. Consequently, large abscissa values ​​do not serve as an approximation for "infinity" here. From a formal theoretical standpoint, the original initial value at an infinite abscissa lies on the boundary of the Lipschitz domain; thus, according to the existence and uniqueness theorem, it falls outside the region of continuity for the right-hand side of the equation. "Unusual" solution behavior is therefore to be expected. Furthermore, if a solution function is obtained via a valid algorithm for an initial or boundary condition at infinity, one must still verify whether the result aligns with the physically plausible outcomes of the reality modeled by the equation.
Regardless of this, investigating the asymptotic behavior and singularities of solutions is, of course, a matter of great theoretical interest.

@sand15 

......using a well-known "classic" method. Aside from the technique involving a ray originating from the point under investigation, the only other method I recall involves the use of angle sums. In this approach, line segments are drawn from the selected point to the vertices of the polygon. The angles lying between adjacent segments are then calculated. If the sum of these angles is less than Pi, the point lies outside the polygon. Interior points yield an angle sum of 2*Pi, while points situated on the boundary yield an angle sum of exactly Pi.

As I have just read in one of your sources, the angle method is described there.

p.s.

Long-established methods from complex analysis (such as Cauchy’s theorems and Laurent series) used to present quite a challenge in practical application. It was only after some pondering over your solution that I realized - just now, in fact - how effectively these methods can yield solutions through programming... provided, of course, that one knows how to do it (which I don't be able :-( ). Back in my student days, the winding number was jokingly referred to in academic circles as a "zero-counter." In Maple, however, this is now a quick, "dry" process. In this respect, your solution has opened up a glimpse into a "new world" for me.

@vv @Rouben Rostamian  

...Your solutions - much like the others published here - is certainly worth keeping. From a programming standpoint, however, I am still far from capable of writing such command sequences myself. In the meantime, though, I have learned to read programs with understanding. Consequently, I can utilize your solutions as building blocks in my own small projects. And regarding the topic of "lattice polygons," there is a multitude of problems available - not least among them those from the IMO.
Thank you very much!

@sand15 

Your reference to literature is welcome. Familiar topics are presented in a methodologically sound and easily understandable way. Apparently, French mathematicians have a special knack for geometry and number theory (Desargue, Fermat, ...) ;-) .

@janhardo 

... is not my native language. Therefore, my translation may have been misunderstood - my apologies. Hence, a new attempt: It is surprising that a program as powerful as Maple lacks such a simple and convenient approach to the subject of lattice points and lattice polygons - unlike, for instance, the program Cabri. As a relatively inexperienced Maple user, I would find this very helpful.

@Rouben Rostamian  

These are exactly the kinds of polygons I want to generate in a straightforward way. Apparently, Maple only offers the option to do this using lists of coordinates. But regardless: Thanks—that’s exactly what I needed. It's a shame that it's not as comfortable as in Cabri.

@nm 

...and how is a grid polygon drawn with pinpoint precision?

@sand15 

After copying your algorithm, the following error message appears:

Error, illegal use of an object as a name

What am I doing wrong?

@Jean-Michel 

...Mr. Peano is to blame ;-) . His final formulation of the axioms named after him defines zero as a natural number. And for high school students and undergraduates, this is entirely sufficient and practically applicable. Naturally, in so-called "higher mathematics" a class-based definition grounded in set theory is employed. However in the otherwise excellent book *Exposé moderne des mathématiques élémentaires* by Lucienne Félix (page 553) zero is omitted during the introduction of the natural numbers, and the sequence begins with 1. It is not until the introduction of the integers that zero necessarily makes its appearance. In this respect, the book reverts to the original form of Peano's axioms.