How quickly can you do this?

Computer Science Level 5

Let $a$ be the number of tuples $(a_1,a_2,a_3,\ldots)$ that satisfy the equation given below : $\sum_{i=1}^{\infty} ia_{i}=45$

Let $b$ be the number of tuples $(b_1,b_2,b_3,\ldots)$ that satisfy the inequality given below : $\sum_{i=1}^{\infty} (i+1)b_{i}\leq 45$

Enter your answer by concatenating $a$ and $b$ . For example, if $a=23$ and $b=65$ , then enter $2365$ as your answer.

Details and Assumptions

$a_{i}\geq 0\ \forall\ i\in \Bbb Z^{+}$
$b_{i}\geq 0\ \forall\ i\in \Bbb Z^{+}$
$\Bbb Z^{+}$ denotes the set of all positive integers.
By title, I mean how quickly your program can do this.

Here are My CS Problems

The answer is 8913489134.

2 solutions

Ronak Agarwal
Oct 25, 2015

Here's mine code in java

class Maths_Calculator {


    public static void main(String[] args) {

    System.out.print(Partition(45,45)) ;
    }

public static int Partition(int i,int j){
int sum=0 ;
if(i<j)
    return Partition(i,i) ;
else
    if(j==0 || j==1)
      return 1 ;
    else
      {for(int k=1 ; k<j+1 ; k=k+1){
           sum=sum+Partition(i-k,k) ;

           }
      return sum ;      
      }

}

}

Pranjal Jain
Apr 7, 2015

Both of the given problems boils down to number of partitions of $45$ .

For example,

Partitions of 1 are $(1)$
Partitions of 2 are $(1+1),(2)$
Partitions of 3 are $(1+1+1),(2+1),(3)$

This can be easily done using Mathematica,

PartitionsP[45]

which gives output as 89134

Just concatenate it twice to get the answer 8913489134

Anyways, here's python program,

def findN():
    sequence = [[0], [1]]
    sums = [0, 1]
    n = 2
    while ( True ):
        array = []
        for i in xrange( 1, n ):
            x = n - i
            if ( x <= i ):
                array.append( sums[ x ] )
            else:
                s = 0
                for j in xrange( 0, i ):
                    s += sequence[ x ][ j ]
                array.append( s )
        array.append( 1 )
        sequence.append( array )
        sums.append( reduce(lambda x, y: x + y, array) )
        if (n == 45):
            print sums[ -1 ]
            break
        n += 1
findN() #It returns 89134

How quickly can you do this?

Here are My CS Problems

The answer is 8913489134.

2 solutions

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