Two ways:

First way (start with 1, add the odd numbers in succession
 each time)

           1
   1+1  =  2
   2+3  =  5
   5+5  = 10
  10+7  = 17
  17+9  = 26
  26+11 = 37
  36+13 = 50
  50+15 = 65

Second way (add 1 to the square of each integer starting with 0

  0²+1 =  0+1 = 1  
  1²+1 =  1+1 = 2
  2²+1 =  4+1 = 5
  3²+1 =  9+1 = 10
  4²+1 = 16+1 = 17
  5²+1 = 25+1 = 26
  6²+1 = 36+1 = 37
  7²+1 = 49+1 = 50
  8²+1 = 64+1 = 65

Fibonacci sequence: (Start with two 1's, add them to get the 3rd term.
Thereafter find the next term by adding the two preceding terms.

          1
          1
  1+1   = 2
  1+2   = 3
  2+3   = 5
  3+5   = 8
  5+8   = 13
  8+13  = 21
 13+21  = 34
 21+34  = 55
 34+55  = 89
 55+89  = 144

Edwin