Linking Binomial to Number Pyramids

[theoneandonly]

I am studying number pyramids. He is a 3-Based Number Pyramid

8
3 5 It is constructec by each two bricks adding
1 2 3 together to make the next brick.

In this pyramid, Base Number=3
Starting Number=1
Finishing Number=8


Here is my formula for working out the Finishing Number for all pyramids with any Base Number.
2to the power of B-1 S + (B-1) 2to the power of B-2

To prove it works, we use it for the pyramid we have here. We come out with 4S+4...and we then realize that using 4S+4 we can get the finishing number for the pyramid!


What I cannot do is link this to binomial formula, but I know it's possible. If anyone knows, please tell me. Thanks a lot, have a nice day.

[NotLKH]

I am studying number pyramids. He is a 3-Based Number Pyramid

8
3 5 It is constructec by each two bricks adding
1 2 3 together to make the next brick.

In this pyramid, Base Number=3
Starting Number=1
Finishing Number=8


Here is my formula for working out the Finishing Number for all pyramids with any Base Number.
2to the power of B-1 S + (B-1) 2to the power of B-2

To prove it works, we use it for the pyramid we have here. We come out with 4S+4...and we then realize that using 4S+4 we can get the finishing number for the pyramid!


What I cannot do is link this to binomial formula, but I know it's possible. If anyone knows, please tell me. Thanks a lot, have a nice day.

Is this your equation:
2^(B-1)*S + (B-1)*2^(B-2)


Question:

When you go from Line 1 2 3 to Line 3 5, you use the last entry of line 1 to start line 2.

Why then does Line 3 not start with 5 but 8?

[theoneandonly]

Yes, it is. I apologize for my poor presentation, if you think you can help, it would be great. Thanks

[NotLKH]

Question:

When you go from Line 1 2 3 to Line 3 5, you use the last entry of line 1 to start line 2.

Why then does Line 3 not start with 5 but 8?

[theoneandonly]

No, no, you don't understand. Line 1 has 3 numbers. 1+2=3, to make the first number of the second line. 2+3=5 to make the second number. In the second line, we only have two number, so it's 3+5=8. And in the third line we have only one number, so that is our finishing number. Can you help? Thanks alot

[NotLKH]

Ohh, I see now.


Given a line whose sequence is 1 2 3 4 5

The next line is 3 5 7 9
And on top of that is 8 12 16

...20 28
...48


Hmmm. I'll think about this.

[theoneandonly]

Yes, that's it. If you could think about linking it to binomial, i'd be very grteful. Thanks.

[NotLKH]

Pardon my ignorance, and I suppose I could google, but perhaps you can briefly describe what the binomial formula is?

[zaza]

The binomial formula describes the expansion of (x+a)^n, i.e. it expresses it in terms of sums of different powers of x and a.

The binomial expansion is the sum from i=0 to n of (n,i)(a^i)(x^[n-i]) where (n,i) is the binomial coefficient.
The binomial coefficient is calculated as n!/i!(n-i)!

When you work these out for different values of n, and all values of i, you get Pascal's Triangle, i.e.


n=0: 1
n=1: 1 1
n=2: 1 2 1
n=3: 1 3 3 1
n=4: 1 4 6 4 1


etc. You can see that in each line the i'th coefficient is the sum of the two numbers above it. This is starting to sound suspiciously like your pyramid already.

[EDIT: Cursed formatting: Write the above in pyramid style, or consider each number as the sum of the one above and the one above-left]



In your example, you have a pyramid of 1 2 3

In the next line, you have 1+2=3 and 2+3=5. So you've counted the 1 once, the 2 twice and the 3 once. Your top line is the sum of all of these, so it is

1 + 2 + 2 + 3 = 8.

There is a link here to the n=2 line of Pascal's Triangle.

To generalise: Try writing down a simple base 3 pyramid using a, b, c as your starting values. What is the line above, and then the line above that in terms of a, b, and c?

Now try it using a, b, c, d for a base 4 pyramid. What does the sum of the top number come out as now?

See the link?


zaza