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Topic: Triangular numbers: 3 questions (Read 3562 times) |
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Benny
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Triangular numbers: 3 questions
« on: Aug 23rd, 2011, 4:52pm » |
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#1
In OEIS A000217 0 is on the list
But in wikipedia and Mathworld 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?
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| « Last Edit: Aug 23rd, 2011, 11:12pm by Benny » |
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towr
wu::riddles Moderator
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Re: Triangular numbers: 3 questions
« Reply #1 on: Aug 24th, 2011, 10:50am » |
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on Aug 23rd, 2011, 4:52pm, BenVitale wrote:| Is zero a triangular number or not? |
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If Tn = 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.
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Wikipedia, Google, Mathworld, Integer sequence DB
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