%I #43 Jul 09 2019 09:08:52
%S 331,233,10177,224,10314,10323,210,203,110,10717,84700,420,121,340,
%T 210,206,236,10182,10454,112,206,99300,10217,10323,420,212,103,326,
%U 10033,136,212,110,206,110,10033,270,117,470,1008224,43400,170,11000,10024,21400,14201,206,410,420,212,236,1004644,10066,224,32100,10043,121
%N Underline the central digit of all terms: the underlined digits reconstruct the starting sequence. This is also true if one translates the sequence in French and underlines the central letter of each word: the underlined letters spell the (French) sequence again. This is the lexicographically earliest sequence where repeated terms are admitted.
%C By construction, all integers have here an odd number of digits and an odd number of letters in their French translation.
%C The method used by the author was first to build a 21 X 10 array (10 digits X 21 letters) and to fill the 210 squares with terms t[$,d] that have the central letter $ and the central digit d.
%C The Graner link explains why the author's array didn't take into account the letters B, J, K, W and Y.
%C The second step was to find a(1). This term had to be the smallest one starting with a digit d and having d in its center, PLUS the French translation of this term had to start with a letter $ and show the same letter $ in its center. This term is 331 (the digit 3 starts the integer and the digit 3 stands in the middle of it PLUS the letter T starts TROIS CENT TRENTE ET UN and the letter T stands in the middle of the word TROIS CENT (T)RENTE ET UN  hyphens and spaces don't count as they are not letters).
%C The rest of the sequence comes now automatically; a(2) for instance must be the smallest term t[R,3] when translated in French. This is [DEUX CENT T(R)ENTETROIS,233].
%C Similarly a(3) must be the smallest term t[O,1] when translated in French and this is [DIX MILLE CENT S(O)IXANTEDIXSEPT,10177], etc.
%C The first term diverging from A319921 is a(21) = [DEUXC(E)NTSIX, 206] and not a(21) = [TROISC(E)NTDEUX, 302] as repeated terms are allowed here.
%C It might be of interest to show the equivalent sequences in other languages (English, German, Spanish, Italian, etc.) [Note: I think the French version is enough!  _N. J. A. Sloane_, Sep 27 2018]
%D Eric Angelini, Maths étonnant, tangente, No. 189, juilletaoût 2019, p. 29.
%H JeanMarc Falcoz, <a href="/A319718/b319718.txt">Table of n, a(n) for n = 1..1001</a>
%H Nicolas Graner, <a href="https://www.graner.net/nicolas/nombres/nom.php">Les grands nombres en français</a> (in French).
%e The sequence starts with 331, 233, 10177, 224, 10314, and the central (underlined) digits are 3,3,1,2,3,... which are precisely the digits starting the sequence itself; now the successive 5 unique central letters of the above 5 French terms are T, R, O, I, S and this spells the beginning of TROIS CENT TRENTE ET UN, the term a(1).
%Y Cf. A319921 (repeated terms are forbidden, in contrast to this sequence).
%K nonn,base,word
%O 1,1
%A _Eric Angelini_ and _JeanMarc Falcoz_, Sep 26 2018
