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A319718
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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.
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4
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331, 233, 10177, 224, 10314, 10323, 210, 203, 110, 10717, 84700, 420, 121, 340, 210, 206, 236, 10182, 10454, 112, 206, 99300, 10217, 10323, 420, 212, 103, 326, 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
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OFFSET
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1,1
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COMMENTS
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By construction, all integers have here an odd number of digits and an odd number of letters in their French translation.
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.
The Graner link explains why the author's array didn't take into account the letters B, J, K, W and Y.
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).
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)ENTE-TROIS,233].
Similarly a(3) must be the smallest term t[O,1] when translated in French and this is [DIX MILLE CENT S(O)IXANTE-DIX-SEPT,10177], etc.
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.
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]
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LINKS
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Eric Angelini, Des suites inouïes, Maths étonnantes, Tangente, No. 189, juillet-août 2019, p. 29.
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EXAMPLE
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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).
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CROSSREFS
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Cf. A319921 (repeated terms are forbidden, in contrast to this sequence).
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KEYWORD
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nonn,base,word
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AUTHOR
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STATUS
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approved
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