Question 639539
<pre>
 X. Y                        Y
 1. 1                    =   1
 2. 1+2                  =   3
 3. 1+2+4                =   7
 4. 1+2+4+8              =  15
 5. 1+2+4+8+16           =  31
 6. 1+2+4+8+16+32        =  63
 7. 1+2+4+8+16+32+64     = 127
 8. 1+2+4+8+16+32+64+128 = 255

The simplified table looks like this:

 X.   Y                        
 1.   1
 2.   3
 3.   7
 4.  15
 5.  31
 6.  63
 7. 127
 8. 255

Notice that each number in the Y column is 1 less than
a power of 2.

 X.   Y                        
 1.   1 = 1 less than   2 = 2<sup>1</sup> - 1
 2.   3 = 1 less than   4 = 2<sup>2</sup> - 1
 3.   7 = 1 less than   8 = 2<sup>3</sup> - 1
 4.  15 = 1 less than  16 = 2<sup>4</sup> - 1
 5.  31 = 1 less than  32 = 2<sup>5</sup> - 1
 6.  63 = 1 less than  63 = 2<sup>6</sup> - 1
 7. 127 = 1 less than 128 = 2<sup>7</sup> - 1
 8. 255 = 1 less than 256 = 2<sup>8</sup> - 1

  I'll remove the middle
two columns to make it clearer:

 X.   Y                        
 1. 2<sup>1</sup> - 1
 2. 2<sup>2</sup> - 1
 3. 2<sup>3</sup> - 1
 4. 2<sup>4</sup> - 1
 5. 2<sup>5</sup> - 1
 6. 2<sup>6</sup> - 1
 7. 2<sup>7</sup> - 1
 8. 2<sup>8</sup> - 1

Observing the pattern, we see that in every case the exponent
of 2 is the same as the value of x.  So that will have to be the
case when x=15, and we can jump to line 15

15. 2<sup>15</sup> - 1

Punching that out on a calculator gives 32767

A general equation that relates all values of x and y is

 y = 2<sup>x</sup> - 1

Edwin</pre>