Triangular number

Abstract: 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 + l) ^

Abstract: Imagine having some marbles, pebbles, or other objects that you want to lay out in a neat triangular pattern. How many do you need to end up with a complete triangle? Three will do; so will 6, 10, 15, and... These numbers are called triangular numbers. Ask the same question for a triangular pyramid. Four will do; so will 10, 20, 35, and... the so-called pyramidal numbers. This book looks at the discovery of the multiplicity of properties and uses triangular numbers and their many extensions possess. Although often displayed in a triangular array named after the seventeenth-century French philosopher Blaise Pascal, triangular numbers were known many centuries earlier. In this book A.W.F. Edwards traces the Arithmetical Triangle back to its roots in Pythagorean arithmetic, Hindu combinatorics, and Arabic algebra, and gives an account of the progressive solution of combinatorial problems from the earliest recorded examples to the work of Renaissance and later mathematicians. He shows how Pascal's work -- so modern in style -- in establishing the properties of the numbers and their application in various fields led to Newton's discovery of the binomial theorem for fractional and negative indices and to Leibniz's discovery of calculus.

Abstract: Studied within the family, the relationship between the number of group members and the numbers of personal relationships between them may be expressed with mathematical precision. The Law of Family Interaction here proposed is that, with the addition of each person to a family of primary group, the number of persons increases in simple arthmetical progression while the number of personal interrelationships within the group increases in the order of triangular numbers. This law may be applied to various problems of family relationships.