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        riddles >> easy >> Triangular numbers: 3 questions
        
(Message started by: BenVitale on Aug 23^rd, 2011, 4:52pm)

Title: Triangular numbers: 3 questions
Post by BenVitale on Aug 23^rd, 2011, 4:52pm

#1

In OEIS A000217 (http://oeis.org/A000217) 0 is on the list
But in wikipedia (http://en.wikipedia.org/wiki/Triangular_number) and Mathworld (http://mathworld.wolfram.com/TriangularNumber.html) 0 is not on the list of triangular numbers.

Is zero a triangular number or not?

#2

We know that a triangular number can never end in 2, 4, 7 or 9
So, 0, 1, 3, 5, 6 and 8 are the possible final digits of a triangular number.

1-digit triangular numbers:
0, 1, 3, 6
4/6 or = 66.6667% of digits

2-digit triangular numbers:
10, 15, 21, 28, 36, 45, 55, 66, 78, 91
0, 1, 5, 6, 8
5/6 or = 83.3333%

With 3-digit triangular numbers, we get 100%

Are there any studies about the repetition or frequency of the last digit of triangular numbers?

EDIT

I'm going to add a third question:

The possible final digits of a triangular number:
0, 1, 3, 5, 6, 8

Instead of considering 1-digit, now I'm asking about the final 2 digits.
For example, is there a triangular number that ends 00? ends 68? 71?
Of the 100 possible 2-digit endings how many actually occur?

Title: Re: Triangular numbers: 3 questions
Post by towr on Aug 24^th, 2011, 10:50am

on 08/23/11 at 16:52:37, BenVitale wrote:

Is zero a triangular number or not?

If T_n = n(n+1)/2, then T₀ = 0. So by that criterion the 0th triangular number is 0. And the -1th is also 0, and the -2th is -1, etc.
Also, if you can fill an equilateral triangle uniformly with 0 dots, so by that criterion 0 is also a triangular number.