VOL. 89, NO. 3, JUNE 2016
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Proof Without Words: Every Cobalancer Is a Balancing Number G. K. P A N D A RAVI KUMAR DAVALA
National Institute of Technology Rourkela,India gkpanda
[email protected] [email protected]
As defined by Panda and Ray [3], a cobalancing number (with cobalancer r ) is a natural number n satisfying 1 + 2 + · · · + n = (n + 1) + · · · + (n + r ). The cobalancing numbers satisfy the recurrence relation bn+1 = 6bn − bn−1 + 2 with initial values b1 = 0, b2 = 2. A natural number n is called a balancing number [1] with balancer r if 1 + 2 + · · · + (n − 1) = (n + 1) + (n + 2) + · · · + (n + r ). Theorem 1. A natural number r is a cobalancer (corresponding to a cobalancing number n) if and only if r is a balancing number [3]. Proof.
Jones [2] showed visually the relationship between balancing numbers and triangular numbers. The following exercise relates balancing and cobalancing numbers. Exercise. Show the following result. A natural number r is a balancer (corresponding to a balancing number n) if and only if r is a cobalancing number [3]. c Mathematical Association of America Math. Mag. 89 (2016) 165–166. doi:10.4169/math.mag.89.3.165. MSC: Primary 11B39; Secondary 11B83
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REFERENCES 1. A. Behera, G. K. Panda, On the square roots of triangular numbers, Fib. Quart. 37 no. 2 (1999) 98–105. 2. M. A. Jones, R. B. Nelsen, Proof without words: The square of a balancing number is a triangular number, College Math. J. (2012) 212. 3. G. K. Panda, P. K. Ray, Cobalancing numbers and cobalancers, Int. J. Math. Math. Sci. 2005 no. 8 (2005) 1189–1200. Summary.
A visual proof that every cobalancer is a balancing number.
GOPAL KRISHNA PANDA (MR Author ID: 328728) is a professor of mathematics at National Institute of Technology, Rourkela, India. Currently, he holds the position of dean of sponsored research, industrial consultancy and continuing education. He has coauthored over 40 publications in the areas of nonlinear optimization and number theory. He coauthored papers giving concepts of balancing and cobalancing numbers. RAVI KUMAR DAVALA (MR Author ID: 1098160) obtained his M.Sc. degree in mathematics from Indian Institute of Technology, Kanpur, India. He is currently a Ph.D. student under the supervision of Prof. Gopal Krishna Panda. His research interest is perfect numbers in binary recurrence and geometrical study of such sequences.
