A319718 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.

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

Offset: 1

Author

Eric Angelini and Jean-Marc Falcoz, Sep 26 2018

Keywords

• nonn
• base
• word

Comments

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]

Examples

			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).
		

Links

• Jean-Marc Falcoz, Table of n, a(n) for n = 1..1001
• Eric Angelini, Des suites inouïes, Maths étonnantes, Tangente, No. 189, juillet-août 2019, p. 29.
• Nicolas Graner, Les grands nombres en français (in French).

Crossrefs

Cf. A319921 (repeated terms are forbidden, in contrast to this sequence).