Question

  for an arbitrary word, how can we derive "i" given the pattern below?

  for a (i = 1)
  a

  for ab (i = 3)
  a
  ab
  b

  for abc (i = 6)
  a
  ab
  abc
  b
  bc
  c

  for abcd (i = 10)
  a
  ab
  abc
  abcd
  b
  bc
  bcd
  c
  cd
  d

  for abcde (i = 15)
  a
  ab
  abc
  abcd
  abcde
  b
  bc
  bcd
  bcde
  c
  cd
  cde
  d
  de
  e