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 A303950 A fractal-like sequence: erasing all pairs of contiguous terms that sum up to a Fibonacci number leaves the sequence unchanged. 6

%I

%S 1,2,4,9,1,3,5,2,4,6,7,9,1,3,8,13,5,2,4,6,10,11,7,9,1,3,8,12,22,13,5,

%T 2,4,6,10,14,20,11,7,9,1,3,8,12,15,19,22,13,5,2,4,6,10,14,16,18,20,11,

%U 7,9,1,3,8,12,15,17,38,19,22,13,5,2,4,6,10,14,16

%N A fractal-like sequence: erasing all pairs of contiguous terms that sum up to a Fibonacci number leaves the sequence unchanged.

%C The sequence is fractal-like as it embeds an infinite number of copies of itself.

%C The sequence was built according to these rules (see, in the Example section, the parenthesization technique):

%C 1) no overlapping pairs of parentheses;

%C 2) always start the content inside a pair of parentheses with the smallest integer F not yet present inside another pair of parentheses;

%C 3) always end the content inside a pair of parentheses with the smallest integer I not yet present inside another pair of parentheses such that the sum F + I is a Fibonacci number;

%C 4) after a(2) = 2, always try to extend the sequence with a duplicate of the oldest term of the sequence not yet duplicated; if this leads to a contradiction, open a new pair of parentheses.

%H Lars Blomberg, <a href="/A303950/b303950.txt">Table of n, a(n) for n = 1..998</a>

%e Parentheses are added around each pair of terms that sum up to a Fibonacci:

%e (1,2), (4,9), 1, (3,5), 2, 4, (6,7), 9, 1, 3, (8,13), 5, 2, 4, 6, (10,11), 7, 9, 1, 3, 8, (12,22), 13, 5, 2, 4, 6, 10, (14,20), 11, ...

%e Erasing all the parenthesized contents yields

%e (...), (...), 1, (...), 2, 4, (...), 9, 1, 3, (....), 5, 2, 4, 6, (.....), 7, 9, 1, 3, 8, (.....), 13, 5, 2, 4, 6, 10, (.....), 11, ...

%e We see that the remaining terms slowly rebuild the starting sequence.

%Y Cf. A000045 (Fibonacci numbers).

%Y For other "erasing criteria", cf. A303845 (prime by concatenation), A274329 (pair summing up to a prime), A303936 (pair not summing up to a prime), A303948 (pair sharing a digit), A302389 (pair having no digit in common).

%K nonn,base

%O 1,2

%A _Eric Angelini_ and _Lars Blomberg_, May 03 2018

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Last modified September 22 11:29 EDT 2020. Contains 337289 sequences. (Running on oeis4.)