### d-pyramidal number?

Mar. 4th, 2009 06:24 pm*n*natural numbers is the

*n*th triangular number. And the sum of the first

*n*triangular numbers is the

*n*th tetrahedral number.

Is there a term that generalizes this concept? Something along the lines of a "

*d*-pyramidal number", where if

*d*= 1 you get the natural numbers, for

*d*= 2 you get triangular numbers, and for

*d*= 3 you get tetrahedral numbers, and so on?

I was trying to sleep last night with the

*Twelve Days of Christmas*going through my head and wondered how many gifts were given at the end of the 12 days. (It's 364.) As I was showering this morning I went through several strategies to derive this answer (along the way, lamenting that if I wasn't dealing with integers then I could just do a quick integral, and that my visualization skills in three dimensions is not very good), and on the bus I ended up coming up with (

*n*+

*d*- 1) choose

*d*as the general answer for the

*n*th

*d*-pyramidal number.

(Actually I just had an uglier product--the pithy version of the formula didn't occur to me until I went online searching for the right terminology and found a site that shows the answer in a Pascal's triangle, and made me go, "duh, what I have can be expressed more concisely!")