Balancing numbers n and balancers r are solutions of the Diophantine equation 1 + 2 + . . . + (n 1) = (n + 1) + (n + 2) + . . . + (n + r). It is well-known that if n is a balancing number, then 8n2 + 1 is a perfect square and its positive square root is called a Lucas-balancing number. In this paper, some new identities involving balancing and Lucas-balancing numbers are obtained. Some divisibi...

1. BALANCING NUMBERS We call an Integer n e Z a balancing number if 1+ 2+ --+ (»l ) = (w + l) + (w + 2) +••• + (» + >•) (1) for some r e Z. Here r is called the balancer corresponding to the balancing number n. For example, 6, 35, and 204 are balancing numbers with balancers 2, 14, and 84, respectively. It follows from (1) that, if n is a balancing number with balancer r, then n2^(n + r)(n + r ...

designing the self-balancing two-wheeled mobile robots and reducing undesired vibrations are of great importance. for this purpose, the majority of researches are focused on application of relatively complex control approaches without improving the robot structure. therefore, in this paper we introduce a new two-wheeled mobile robot which, despite its relative simple structure, fulfills the req...

Pseudo-splines are a rich family of functions that allows the user to meet various demands for balancing polynomial reproduction (i.e., approximation power), regularity and support size. Such a family includes, as special members, B-spline functions, universally known for their usefulness in different fields of application. When replacing polynomial reproduction by exponential polynomial reprod...

The advancement of information and communication technologies such as personal computers, the Internet and mobile phones has enabled people to work any time and anywhere. Teleworking, the practice of setting up home offices for employees with appropriate resources for computing and communication, is one example of this new flexibility. Teleworking brings new challenges as well as benefits, and ...

Load Balancing is the standout amongst the most noteworthy parts in circulated situations. As Cloud Computing is outstanding amongst other stage that gives stockpiling of information in extremely insignificant cost and available forever finished the internet, load balancing for the cloud computing has transformed into an exceptionally intriguing and imperative investigation region. Load balanci...

Acoustic waves existing in hypersonic boundary layers act as a heat pump that transfers energy from the sonic line to wall causing temperature rise, which explains newly identified aerodynamic heating related Mack's second mode perspective of thermoacoustic effects. The analysis data direct numerical simulations shows mode, belonging family trapped acoustic waves, is highly amplified Mach 6 lay...