Comments on: Pascal’s Triangle

By: Janaka

Janaka — Sat, 08 Oct 2022 12:08:43 +0000

It says when the first number right to 1 is prime, it divides all in the raw , but it should be all except first and last numbers in the raw.

By: Brandon Busby

Brandon Busby — Thu, 20 Aug 2020 20:38:42 +0000

I am working on the following problem. I realized that the underlying structure IS the Fibonacci sequence. In order to solve the problem, I need a way to compute the diagonals shown above in a computationally efficient way.

I.e., I need a way to efficiently compute the following sequences:
– 1
– 1 1
– 1 2
– 1 3 1
– 1 4 3
– 1 5 6 1
– 1 6 10 4
– 1 7 15 10 1
– …

Ideally, to compute the nth sequence would require time proportional to n. One way that this could be achieved is by using the (n-1)th sequence to compute the nth sequence.

If there happens to be a way to compute the nth sequence in constant time, that would be fantastic.

Problem:

There is a fence with n posts, each post can be painted with one of the k colors.

You have to paint all the posts such that no more than two adjacent fence posts have the same color.

Return the total number of ways you can paint the fence.

Note:
n and k are non-negative integers.

Example:

Input: n = 3, k = 2
Output: 6
Explanation: Take c1 as color 1, c2 as color 2. All possible ways are:

post1 post2 post3
—– —– —– —–
1 c1 c1 c2
2 c1 c2 c1
3 c1 c2 c2
4 c2 c1 c1
5 c2 c1 c2
6 c2 c2 c1

By: Callum

Callum — Tue, 03 Dec 2019 14:34:41 +0000

In reply to Gary Meisner.

It also works below the 5th line. You just carry the tens digit into the previous column

****11^5=161051 is different than 15101051***
1,5,10,10,5,1
1(5+1)(0+1)051
1(6)(1)051

By: M

Fri, 31 May 2019 10:51:08 +0000

In reply to Finn.

Looking at it this way might help:

1
0 1
1 0 1
0 2 0 1
1 0 3 0 1
0 3 0 4 0 1
1 0 6 0 5 0 1

Row sum = Fibonacci sequence

By: Thomas

Thomas — Thu, 09 May 2019 03:23:44 +0000

I was trying to find the fibonacci sequence in the pascal’s triangle. This website is so useful!!!

By: Mark

Mark — Sun, 31 Mar 2019 09:57:19 +0000

In reply to Mark.

An inverse example:

(1x) 0 =
(1x) 1 + (1x) -1 =
(1x) -1 + (2x) 2 + (1x) -3 =
(1x) 2 + (3x) -3 + (3x) 5 + (1x) -8 =
(1x) -3 + (4x) 5 + (6x) -8 + (4x) 13 + (1x) -21 = 0

By: Gary B Meisner

Gary B Meisner — Sat, 29 Dec 2018 11:01:24 +0000

In reply to Finn.

See the illustration. The green lines are the “diagonals” and the numbers of the Pascal’s triangle they intersect sum to form the numbers of the Fibonacci sequence – 1, 1, 2, 3, 5, 8, …

By: Finn

Finn — Fri, 21 Dec 2018 05:54:06 +0000

So I don’t understand. What does it mean when it says “the numbers on the diagonals add to the Fibonacci series”. Which diagonals is this referring to, and how does this add to make the sequence?

By: Mark

Mark — Thu, 30 Aug 2018 13:41:50 +0000

Another application:
(1x) 21 =
(1x) 8 + (1x) 13 =
(1x) 3 + (2x) 5 + (1x) 8 =
(1x) 1 + (3x) 2 + (3x) 3 + (1x) 5 =
(1x) 0 + (4x) 1 + (6x) 1 + (4x) 2, (1x) 3 = 21

By: Mark

Mark — Tue, 01 May 2018 08:31:26 +0000

Althought known as Pascal’s triangle, apparently Pascal himself wrote it as a square. As a square rows and columns represent negative powers of 9 (10-1).
1 1 1 1 1 1
1 2 3 4 5
1 3 6 10
1 4 10
1 5
1

1/9 = 0,1111111
1/81=0,0123456
1/729= 0.00137
etc.
(using 1/99…. will avoid carrying over of decimals)

Addiing up those fractions ‘aproaches’ the ratio 1/8 = 0,125 (0,1249999999…..)
Similar the infinite sum of negative powers of 90 (1/90) results in 1/89, which decimally represents the diagonal sum of Pascal’s triangle:
1 1 1 1 1 …
0 0 1 2 3 4 …
0 0 0 0 1 3 6 …
0 0 0 0 0 0 1 4 …
0 0 0 0 0 0 0 0 1 …
—————————— +
1 1 2 3 5 …