A319921
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 of distinct terms.
Original entry on oeis.org
331, 233, 10177, 224, 10314, 10323, 210, 203, 110, 10717, 84700, 420, 121, 340, 311, 206, 236, 10182, 10454, 112, 302, 99300, 10217, 10331, 10206, 212, 103, 326, 10033, 136, 216, 217, 305, 218, 10084, 270, 117, 470, 1008224, 43400, 170, 11000, 10024, 21400, 14201, 307, 410, 10210, 313, 332, 1004644, 10066, 10304, 32100, 10184, 122
Offset: 1
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).
The first term diverging from A319718 is a(21) = [TROISC(E)NTDEUX, 302] as a(21) is the smallest integer > a(16) = [DEUXC(E)NTSIX, 206], both having a central (underlined) letter E.
- 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. This link explains why the authors didn't take into account the letters B, J, K, W and Y.
Cf.
A319718 (repeated terms are allowed, in contrast to this sequence)
A365705
Underline the digit immediately to the right of the center of each term (see the Comments section for the definition of "center"). This is the lexicographically earliest sequence of distinct integers > 9 such that the successive underlined digits duplicate the sequence itself, digit by digit.
Original entry on oeis.org
11, 21, 12, 31, 41, 22, 13, 51, 14, 61, 32, 42, 71, 23, 15, 81, 91, 24, 16, 101, 33, 52, 34, 62, 17, 111, 72, 43, 121, 25, 18, 131, 19, 141, 82, 44, 151, 26, 161, 10, 171, 53, 63, 35, 92, 73, 54, 36, 102, 181, 27, 191, 201, 211, 37, 112, 64, 83, 221, 122, 231, 132, 45, 241, 28, 251, 93
Offset: 1
The first twelve terms of the sequence are:
11, 21, 12, 31, 41, 22, 13, 51, 14, 61, 32, 42.
We put parentheses around the digit right of center:
1(1), 2(1), 1(2), 3(1), 4(1), 2(2), 1(3), 5(1), 1(4), 6(1), 3(2), 4(2).
The twelve digits in parentheses are:
1, 1, 2, 1, 1, 2, 3, 1, 4, 1, 2, 2.
The above twelve digits are the same as the first twelve digits of the sequence:
11, 21, 12, 31, 41, 22.
-
a[1]=11;a[n_]:=a[n]=(k=10;While[MemberQ[ar=Array[a,n-1],k]||IntegerDigits[k][[Ceiling[IntegerLength@k/2]+1]]!=Flatten[Join[Flatten[IntegerDigits/@ar],IntegerDigits@k]][[n]],k++];k);Array[a,70] (* Giorgos Kalogeropoulos, Sep 21 2023 *)
A365704
Underline the digit immediately to the left of the center of each term (see the Comments section for the definition of "center"). This is the lexicographically earliest sequence of distinct integers > 9 such that the successive underlined digits duplicate the sequence itself, digit by digit.
Original entry on oeis.org
10, 1000, 11, 1001, 1002, 1003, 12, 13, 14, 1004, 1005, 15, 16, 1006, 1007, 20, 17, 1008, 1009, 30, 18, 21, 19, 31, 100, 40, 101, 1010, 1011, 41, 102, 1012, 1013, 50, 103, 51, 104, 60, 105, 1014, 1015, 61, 106, 1016, 1017, 70, 22, 1018, 107, 71, 108, 1019, 1020, 80, 109, 1021, 1022, 90, 32, 1023, 110, 81, 23
Offset: 1
The first twelve terms of the sequence are:
10, 1000, 11, 1001, 1002, 1003, 12, 13, 14, 1004, 1005, 15.
We put parentheses around the digit left of center:
(1)0, 1(0)00, (1)1, 1(0)01, 1(0)02, 1(0)03, (1)2, (1)3, (1)4, 1(0)04, 1(0)05, (1)5.
The twelve digits in parentheses are:
1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1.
The above twelve digits are the same as the first twelve digits of the sequence:
10, 1000, 11, 1001.
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a[1]=10;a[n_]:=a[n]=(k=10;While[MemberQ[ar=Array[a,n-1],k]||IntegerDigits[k][[Floor[IntegerLength@k/2]]]!=Flatten[Join[Flatten[IntegerDigits/@ar],IntegerDigits@k]][[n]],k++];k);Array[a,70] (* Giorgos Kalogeropoulos, Sep 21 2023 *)
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