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A306590 An irregular fractal sequence: underline all terms that follow a comma separating two digits of opposite parities. All underlined terms rebuild the starting sequence. (See the Comments section for more details.) 1

%I

%S 1,3,5,7,9,10,1,11,12,3,13,14,5,15,16,7,17,18,9,19,30,10,1,31,32,11,

%T 33,34,12,3,35,36,13,37,38,14,5,39,50,15,51,52,16,7,53,54,17,55,56,18,

%U 9,57,58,19,59,70,30,10,1,71,72,31,73,74,32,11,75,76,33,77,78,34,12,3,79,90,35,91,92,36,13,93,94,37,95,96,38,14,5

%N An irregular fractal sequence: underline all terms that follow a comma separating two digits of opposite parities. All underlined terms rebuild the starting sequence. (See the Comments section for more details.)

%C The sequence S starts with a(1) = 1 and a(2) = 3. S is extended by duplicating the first term A among the not yet duplicated terms, under the condition that the first digit of A and the last digit of the sequence have opposite parities. If this is not the case for A, we then extend the sequence with the smallest integer X not yet present in S whose first digit has the same parity as the last digit of the sequence.

%H Jean-Marc Falcoz, <a href="/A306590/b306590.txt">Table of n, a(n) for n = 1..20002</a>

%e After a(1) = 1 and a(2) = 3 we cannot have a(3) = 2 as this 2 would be the first underlined term, which is forbidden (the first underlined term of the sequence must be 1 as we want the underlined terms to rebuild the starting sequence). The same argument forbids a(3) = 4 (the digit 4 and the preceding digit 3, being of opposite parities, would force the term 4 to be the first one to be underlined).

%e This procedure forces the first 6 terms to be 1, 3, 5, 7, 9 and 10. But now something has changed: the last digit of the sequence is even. We must then duplicate the first term not yet duplicated, which is 1, as this digit 1 and the 0 of 10 have opposite parities. We thus have a(7) = 1, our genuine first underlined term. In the same spirit follow a(8) = 11 and a(9) = 12 which, again, ends the sequence with an even digit, allowing us to duplicate 3: a(10) = 3.

%e Etc.

%Y Cf. A306580 where the same idea is involved.

%K base,nonn,look

%O 1,2

%A _Eric Angelini_ and _Jean-Marc Falcoz_, Feb 26 2019

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Last modified October 25 09:15 EDT 2021. Contains 348239 sequences. (Running on oeis4.)