[seqfan] Cumulative quantity of letters used in a list of English number-names

[Eric Angelini]

Hello SeqFans,

- this is a sequence S based on English number-names;
- the sequence S is the first column on the left;
- the second column is the quantity of capital 
  letters used so far to spell the terms of the 
  sequence;
- S is always extended with the smallest English 
  number-name that produces a smaller term on the 
  second column than the one on its left.

  S     Cumulative sum of letters used so far
  |     |
  5     4    FIVE
  9     8    NINE
 16    15    SIXTEEN
 26    24    TWENTY-SIX
 35    34    THIRTY-FIVE
 40    39    FORTY
 49    48    FORTY-NINE
 59    57    FIFTY-NINE
 68    67    SIXTY-EIGHT
 79    78    SEVENTY-NINE
 89    88    EIGHTY-NINE
 99    98    NINETY-NINE
116   115    ONE HUNDRED SIXTEEN
136   134    ONE HUNDRED THIRTY-SIX
150   149    ONE HUNDRED FIFTY
169   168    ONE HUNDRED SIXTY-NINE
189   188    ONE HUNDRED EIGHTY-NINE
200   198    TWO HUNDRED
218   216    TWO HUNDRED EIGHTEEN
...

- We see that we cannot start S with ONE as "ONE" has
3 letters, which is not smaller than "1";
- We cannot start S with TWO as "TWO" has 3 letters
which is not smaller than "2";
- We cannot start S with THREE either because "THREE"
has 5 letters which is not smaller than "3";
- We still cannot start S with FOUR because "FOUR" has 
4 letters which is not strictly smaller than "4";
- But FIVE is ok because "FIVE" has 4 letters which is
smaller than "5";
- We cannot extend S with "SIX" as the cumulative quantity
of written letters so far would be 7, which is not smaller
than "6";
- We cannot extend S with "SEVEN" as the cumulative quantity
of written letters so far would be 9, which is not smaller
than "7";
- We cannot extend S with "EIGHT" as the cumulative quantity
of written letters so far would be 9, which is not smaller
than "8";

- But "NINE" is ok because the cumulative quantity of written 
letters so far (NINE included) is 8, which is smaller
than "9";
- If we try now to extend S with TEN, ELEVEN, TWELVE, 
THIRTEEN or FOURTEEN we will always exceed 10, 11, 12, 13
or 14 letters used so far (which is not allowed). If we try
to extend S with FIFTEEN we will see that "FIVE NINE FIFTEEN"
uses 15 letters which is forbidden too. 
But SIXTEEN is ok because "FIVE NINE SIXTEEN" uses 15 letters,
which is less than "16". Etc.

I hope this is of interest for the OEIS and does not contain
too many typos.

Best,
É.