1st row : 1
2nd row : 1 1
3rd row : 1 2 1
4th row : 1 3 3 1
...

nth row : 1 ... ... ... ... 1


There are two 1's per row, except the first row

Number of 1's in n rows = n x 2 - 1 = 2n - 1


1st row : 1 number
2nd row : 2 numbers
3rd row : 3 numbers
...
...
nth row : n numbers


Total number of numbers in n rows

= 1 + 2 + 3 + 4 + 5 + 6 + ... + (n-2) + (n-1) + n

Notice that :

n + 1 = n + 1

2 + (n - 1) = n + 1

3 + (n - 2) = n + 1

You can pair up the numbers this way

Number of pairs = ½n

So sum = (total per pair) x number of pairs

= ½n(n + 1)

Number of integers which are not 1's

= total number of integers - number of 1's

= ½n(n + 1) - (2n - 1)


Required expression

(½n(n + 1) - (2n - 1)) / (2n - 1)

= ½n(n + 1)/(2n - 1) - 1

= n(n+1)/(4n -2) - 1

= (n² + n)/(4n - 2) - 1