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 A306806 An irregular fractal sequence: underline a(n) iff [a(n-1) + a(n)] is divisible by the rightmost digit of a(n); all underlined terms rebuild the starting sequence. 1

%I

%S 1,2,1,3,4,2,1,5,6,3,7,8,4,2,1,9,12,13,14,15,5,16,17,18,6,3,19,22,24,

%T 23,25,26,27,28,7,32,8,4,2,1,29,33,34,35,36,9,37,38,12,39,42,43,44,13,

%U 45,46,14,47,48,49,52,54,53,55,15,5,56,16,57

%N An irregular fractal sequence: underline a(n) iff [a(n-1) + a(n)] is divisible by the rightmost digit of a(n); all underlined terms rebuild the starting sequence.

%C The sequence S starts with a(1) = 1 and a(2) = 2. No term is allowed to end with a 0 digit. S is extended by duplicating the first term A among the not yet duplicated terms, under the condition that the sum [a(n-1) + a(n)] is divisible by the rightmost digit of a(n). If this is not the case, we then extend S with the smallest integer X not yet present in S such that the sum [a(n-1) + a(n)] is not divisible by the rightmost digit of a(n). S is the lexicographically earliest sequence with this property.

%H Jean-Marc Falcoz, <a href="/A306806/b306806.txt">Table of n, a(n) for n = 1..10002</a>

%e S starts with a(1) = 1 and a(2) = 2

%e Can we duplicate a(1) to form a(3)? Yes, as the sum [a(2) + a(3) = 3] is divisible by 1, the rightmost digit of a(3); thus a(3) = 1.

%e Can we duplicate a(2) to form a(4)? No, as the sum [a(3) + a(4) = 3] is not divisible by 2, the rightmost digit of a(4); we thus extend S with the smallest integer X not yet in S such that the sum [a(3) + X] is not divisible by the rightmost digit of a(4); thus a(4) = 3.

%e Can we duplicate a(2) to form a(5)? No, as the sum [a(4) + a(5) = 7] is not divisible by 2, the rightmost digit of a(5); we thus extend S with the smallest integer X not yet in S such that the sum [a(4) + X] is not divisible by the rightmost digit of a(5); thus a(5) = 4.

%e Can we duplicate a(2) to form a(6)? Yes, as the sum [a(5) + a(6) = 6 is divisible by 2, the rightmost digit of a(6); thus a(6) = 2.

%e Can we duplicate a(3) to form a(7)? Yes, as 1 can always be duplicated; thus a(7) = 1.

%e Can we duplicate a(4) to form a(8)? No, as the sum [a(7) + a(8) = 5] is not divisible by 2, the rightmost digit of a(8); we thus extend S with the smallest integer X not yet in S such that the sum [a(7) + X] is not divisible by the rightmost digit of a(8); thus a(8) = 5.

%e Can we duplicate a(4) to form a(9)? No, as the sum [a(8) + a(9) = 8] is not divisible by 3, the rightmost digit of a(9); we thus extend S with the smallest integer X not yet in S such that the sum [a(8) + X] is not divisible by the rightmost digit of a(9); thus a(9) = 6.

%e Can we duplicate a(4) to form a(10)? Yes, as the sum [a(9) + a(10) = 9] is divisible by 3, the rightmost digit of a(10); thus a(10) = 3.

%e Etc.

%Y Cf. A306805 [obtained by replacing the word _rightmost_ by _leftmost_ in the definition. This sequence diverges from A306805 with a(17) = 12, as opposite to a(17) = 20].

%K base,nonn

%O 1,2

%A _Eric Angelini_ and _Jean-Marc Falcoz_, Mar 11 2019

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Last modified May 9 11:42 EDT 2021. Contains 343740 sequences. (Running on oeis4.)