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Lexicographically first sequence starting with a(1) = 1, with no duplicate term, such that a(n) is the result of a self-additive linear combination of its own digits (concatenated sometimes into substrings).
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%I #8 Dec 01 2019 22:17:48

%S 1,100,101,143,182,273,364,429,455,546,637,693,728,819,924,1010,1020,

%T 1030,1040,1050,1060,1070,1080,1090,1233,1370,1371,3288,8833,10000,

%U 10001,10100,10110,10120,10130,10140,10150,10160,10170,10180,10190,10200,10210,10220,10230,10240,10250,10260,10270,10280,10290,10300,10310,10320

%N Lexicographically first sequence starting with a(1) = 1, with no duplicate term, such that a(n) is the result of a self-additive linear combination of its own digits (concatenated sometimes into substrings).

%C A linear combination is an operation like a*u + b*v + c*w + d*x + ... = N. [We want only (+) signs here, thus the word "additive" in the definition.] The coefficients a, b, c, d,... and u, v, w, x,... are determined by the digits of a(n) itself, concatenated sometimes into substrings (no substring with a leading zero is allowed). No digit belongs to more than one substring and all digits are involved in the linear combination.

%C Some patterns are distinguishable:

%C a(1067) = 138614 --- 13 86 14

%C a(1068) = 148515 --- 14 85 15

%C a(1069) = 158416 --- 15 84 16

%C a(1070) = 168317 --- 16 83 17

%C a(1071) = 178218 --- 17 82 18

%C a(1072) = 188119 --- 18 81 19

%C a(1073) = 198020 --- 19 80 20

%C a(1074) = 207921 --- 20 79 21

%C a(1075) = 217822 --- 21 78 22

%C [pattern stops there]

%H Jean-Marc Falcoz, <a href="/A323823/b323823.txt">Table of n, a(n) for n = 1..1165</a>

%e a(1) = 1 belongs to the sequence as 1 = 1*1 [the digits appearing to the left of the (*) sign rebuild a(n); the same is true with the digits appearing to the right of the (*) sign];

%e 10 does not belong to the sequence as 10 = 10*1 + 0*0, although being true, involves 5 digits instead of 4;

%e 36 does not belong to the sequence as 36 = 3*6 + 6*3, although being true and involving 4 digits, doesn't respect the order in which the digits should appear in the linear combination;

%e a(2) = 100 belongs to the sequence as 100 = 10*10 + 0*0;

%e a(3) = 101 belongs to the sequence as 101 = 10*10 + 1*1;

%e a(4) = 143 belongs to the sequence as 143 = 14*1 + 3*43;

%e a(5) = 182 belongs to the sequence as 182 = 18*1 + 2*82;

%e etc.

%Y Cf. A323821 and A323822 for sequences dealing with the same idea of linear combination.

%K base,nonn

%O 1,2

%A _Eric Angelini_ and _Jean-Marc Falcoz_, Jan 30 2019