Dive into the ocean of Combinatorial Patterns and Sequences
About us
Let's Get Acquainted
In the vast realm of mathematics, you'll find a treasure trove of exquisite patterns. Thousands of elegant combinatorial results, ideas, and conjectures have surfaced, primarily centered around triangular numbers, tetrahedral numbers, polygonal numbers, Fibonacci and Lucas numbers, and a myriad of intriguing concepts in elementary number theory. These discoveries and new sequences provide a fascinating glimpse into the harmonious interplay of numbers, showcasing simplicity within their seemingly intricate structures.
Our mathematical journey is an open invitation to mathematicians worldwide. We present these results as an opportunity for exploration, validation, and even friendly challenges. Many of our findings and sequences have resonated within the esteemed N. J. A. Sloane's Online Encyclopedia of Integer Sequences (OEIS), extending their influence into realms like graph theory and beyond. The invaluable resource of N. J. A. Sloane's OEIS has not only guided us but also paved fresh avenues in our mathematical pursuit. Readers associated with OEIS can contribute new comments to further enhance these relevant sequences.
The Online Encyclopedia of Integer Sequences® (OEIS®)
Join us in unraveling mathematical enigmas, guided by the expertise of Dr. K B Subramaniam and fueled by the passion of Amarnath Murthy. Together, we embark on a mesmerizing journey to uncover the inherent beauty and complexity of numbers, patterns within patterns , and the boundless mysteries that await discovery.
Before delving into our results, we encourage you to acquaint yourself with some new definitions and notations that may take on unique meanings within this context. Your suggestions and corrections are warmly welcomed, as some terms may benefit from redefinition and the use of notations and symbols uncommon in other mathematical contexts. As we are dealing only with integers, for convenience we have used (.) instead of (*) to denote the product of two numbers. 4.6 denotes 4*6 = 24.
Let's embark on this mathematical odyssey and explore the captivating world of numbers and sequences together!
Some new terms and notations have been defined in the sample results section below. I would recommend going over the sample results before proceeding further.
There are many more such definitions. Any clash with already defined terms/notations/symbols can be corrected as and when pointed out by readers/experts.
There may be typographical errors (commission/omission), due to copy paste of similar expressions needing only slight modifications. There can be errors of offset. Like in a summation (Σ) , n should have been (n + 1) or (n – 1) or vice versa. The chances of a visible pattern by mere coincidence and subsequent result being wrong may be one in 100. Some of the results might be rediscoveries too, more probably in the case of the ones involving Fibonacci and Lucas numbers. Our apologies for the same.
The flooding of ideas prevented us from proving the results before putting into paper the next one. Huge amount of calculations have been involved in speculating the naturally occurring patterns and subsequent verification for more values of n.
The following websites /applications were quite handy in speeding up the work and in the verification of the results/patterns.
Factoring calculator
https://www.mathpapa.com/factoring-calculator/
5-8 variable equation solver
https://www.handymath.com/cgi-bin/matrix5.cgi
The results should be verified for large values of n before any attempt to establish/prove them analytically. Attempt has been made to find out the pattern within pattern and speculate generalised results by burning midnight oil involving voluminous calculations for over a year. The results and ideas are compiled as and when they occurred and are not organised appropriately. However the results are grouped in sets of 100 each for convenience. There are 22 such sets and 3 more sets are ready to be added shortly making the count to 2500. A rearrangement /reorganization shall be incorporated in due course of time based on feedback .It has been made public prematurely so that more mathematical minds/young minds with access and expertise in computer programs/applications can be applied to explore and establish the volume of combinatorial results.
Most of the results can be established by applying mathematical induction though the process may be clumsy and heavy in algebra. But there are some real tough ones, a challenge for new age mathematicians. The beauty in the patterns shall be evident as one goes through the content. There is a lot of food for thought. We are sure and looking forward to practical useful applications of these results in other fields as it has been a trend that no idea/result in mathematics however abstract has gone waste without application. An evident example is the abstract idea of prime numbers/prime factorisation etc. which is now the backbone of cryptography /data security.
“The deeper one goes into a subject the closer one comes to know of his ignorance”
