text: The generating function for these numbers is x over quantity one minus x, cubed.
full_question: The generating function for these numbers is x over quantity one minus x, cubed. Gauss's "Eureka Theo- rem" is about expressing numbers as sums of three of these numbers. The sum of the first n cubes is the square of one of these numbers. All even perfect numbers are this type of number that can be written as a function of a Mersenne prime. The number of edges in a complete graph is one of these numbers. By looking at a geometric representation of these numbers, one sees that the sum of two consecutive ones is a perfect square. Numbers of this form can be written as n plus 1 choose 2, or n times n plus 1 over 2. For 10 points, iden- tifythesenumberslike15,whichcanbewrittenas1+2+3+4+5.
first_sentence: The generating function for these numbers is x over quantity one minus x, cubed.
raw_answer: triangular numbers [or n plus 1 choose 2 or n times n plus 1 over 2 until they are read]