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while going through the words, I found the following words have some interesting property and hence thought of what could be the next word in the series:

A, ILL, COCOON, _____ ?

After thinking for a little (no...not exactly...after a long) while, I came to know that multiple answers are possible. What can be they and how ?

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    $\begingroup$ This does not seem to have enough data points to be uniquely definable. I can think of a few possibilities on how to complete the sequence, and there's no way to tell which one of them is correct. $\endgroup$ – ffao Jul 12 '17 at 10:46
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    $\begingroup$ Well, @ffao, you can write down your few possibilities and explain how they fit in the series. That could turn to be a correct answer ! Good luck :-) $\endgroup$ – Mea Culpa Nay Jul 12 '17 at 12:08
  • $\begingroup$ LESSEE also fits in place of cocoon. $\endgroup$ – feelinferrety Aug 14 '17 at 14:00
  • $\begingroup$ @feelinferrety... that is correct! $\endgroup$ – Mea Culpa Nay Aug 14 '17 at 14:02
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Here is the recipe for the next word:

A word length of which is 10 and uses only 4 letters (in mixed order), where one of the letters is used once, next is used twice, another is used three times, and the last is used 4 times.

Explanation:

Notice that the first word "A" word is just a word with single letter.
The second word - "ILL" is a word which is consisted of two letters, and one of them ("I") is used once, another one ("L") is used twice.
The third word - "COCOON" is consisted of three letters, one of which ("N") is used once, another is used twice ("C"), and the other ("O") is used three times.
So the next word should be a word which uses 4 letters, where one of the letters is used once, next is used twice, another is used three times, and the last is used 4 times.
Here is the list of the 10 letter words, where you can search the word

UPDATE:
One of the possible words found by @HughMeyers

SLEEVELESS

Also another pack of words, thanks to @Sp3000:

DEADHEADED PEPPERTREE POOPSCOOPS REMEMBERER SERENENESS SLEEVELESS SUSSARARAS SUSURRUSES TERREVERTE

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    $\begingroup$ "sleeveless" springs to mind. Can't think of others off the bat. Not quite interested enough to write a program. $\endgroup$ – Hugh Meyers Jul 12 '17 at 11:35
  • $\begingroup$ @HughMeyers, perfect! I wanted to write a simple program to find such word, but you've found it :) $\endgroup$ – Just Shadow Jul 12 '17 at 11:37
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    $\begingroup$ Qat's pretty useful for this: quinapalus.com/cgi-bin/… $\endgroup$ – Sp3000 Jul 12 '17 at 11:55
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    $\begingroup$ Yes, Just Shadow, you got it correct ! Kudos !! Another such word fits the bill is 'RESTRESSES'. $\endgroup$ – Mea Culpa Nay Jul 12 '17 at 13:15
  • $\begingroup$ These are called "Pyramid Words" in Making The Alphabet Dance by Ross Eckler, which mentions SLEEVELESS as the longest. It doesn't mention the others listed here, but SLEEVELESS does stand out as being rather more common than the rest :) $\endgroup$ – Personman Jul 12 '17 at 18:11

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