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# MY CONTRIBUTIONS

## New concept in Combinatorial Geometry

- Partitioning the set of vertices of a convex polygon into non-intersecting polygons.

## Sequences published on Online Encyclopedia of Integer Sequences (OEIS)

- A350116 – Number of ways to partition the set of vertices of a convex (n +8) – gon into 3 nonintersecting polygons.
- A350248 – Triangle read by rows T(n,k) is the number of noncrossing partitions of an n – set into k blocks of size 3 or more, n > = 0, 0 < = k < = floor (n/3)
- A350286 – Number of different ways to partition the set of vertices of a convex (n + 11) – gon into 4 nonintersecting polygons.
- A350303 – Number of ways to partition the set of vertices of a convex (n + 14) – gon into 5 nonintersecting polygons.
- A350599 – Number of ways to partition the set of vertices of a convex n – gon into nonintersecting directed polygons.
- A350640 – Minimum lcm of the part sizes of a partition of n into parts of size 3 or more.
- A347862 – Total number of polygons left out in all partitions of the set of vertices of a convex n – gon into nonintersecting polygons.
- A351103 – Total number of polygons left over with maximum number of sides when partitioning the set of vertices of a convex n – gon into nonintersecting polygons.
- A352477 – Number of different ways to partition the set of vertices of a convex n – gon into 4 intersecting polygons.
- A352474 – Number of different ways to partition the set of vertices of a convex n – gon into 3 intersecting polygons.
- A352611 – Number of different ways to partition the set of vertices of a convex n – gon into 5 polygons.
- A352900 – Number of different ways to partition the set of vertices of a convex n – gon into intersecting polygons.
- A357603 – Number of different pairs of shortest paths in an n X n lattice going between opposite corners in opposite direction and not meeting at their middle point.
- A357760 – Number of different pairs of shortest grid paths joining two opposite corners in opposite order in an n X n X n grid with middle point on the paths as a common point.
- A358481 – Number of different pairs of shortest grid paths joining two opposite corners in opposite order in an n X n X n grid without having middle point on their paths as a common point.
- A362207 – Number of unordered triples of shortest nonintersecting grid paths joining two opposite corners of an n X n X n grid.
- A360444 – Number of ways for two nonintersecting, unordered pairs of shortest grid paths to cross over between two opposite corners in an n X n grid without intersecting opposite paths at their middle points.
- Mathematical Model of the sequence A360444.
- Python Code of the Mathematical Model of A360444.
- Mathematical Model of ordered triplets ( An open challenge to create a computer program and obtain integer outputs)

## Mathematical Models and Computer Programs in Combinatorial Geometry

- Mathematical Model of the sequence A360444.
- Python Code of the Mathematical Model of A360444.
- Mathematical Model of ordered triplets ( An open challenge to create a computer program and obtain integer outputs)

## New proofs and totally different approaches in other fields of mathematics

22. Geometrical proof of the sum of infinite series Cot^{-1} 3 + Cot^{-1} 7 + Cot^{-1} 13 + …….. converges to π /4.

23. Geometrical proof of the sum of infinite series Cot^{-1} 2 + Cot^{-1} 8 + Cot^{-1} 18 + …….. converges to π /4.

24. Geometrical proof of the sum of infinite series 1/1.2 + 1/2.3 + 1/3.4 + …… converges to 1.

25. Geometrical proof of sum and the product of three tangents of a triangle are equal in the case of an obtuse-angled triangles.

26. The Inclusion-Exclusion Principle in geometrical approach of completing square in Quadratics.

27. Fully trigonometric proof of Pythagoras theorem without applying properties of similar triangles.

28. Simultaneous Single Proof of the Sine Law and Cosine Law.

## 21^{st} Century Teaching and Learning Mathematics

29. Humanistic Mathematics Activity-Based Project.