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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.

%I A303950 #12 Dec 02 2019 04:14:34
%S A303950 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 A303950 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 A303950 7,9,1,3,8,12,15,17,38,19,22,13,5,2,4,6,10,14,16
%N A303950 A fractal-like sequence: erasing all pairs of contiguous terms that sum up to a Fibonacci number leaves the sequence unchanged.
%C A303950 The sequence is fractal-like as it embeds an infinite number of copies of itself.
%C A303950 The sequence was built according to these rules (see, in the Example section, the parenthesization technique):
%C A303950   1) no overlapping pairs of parentheses;
%C A303950   2) always start the content inside a pair of parentheses with the smallest integer F not yet present inside another pair of parentheses;
%C A303950   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 A303950   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 A303950 Lars Blomberg, <a href="/A303950/b303950.txt">Table of n, a(n) for n = 1..998</a>
%e A303950 Parentheses are added around each pair of terms that sum up to a Fibonacci:
%e A303950 (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 A303950 Erasing all the parenthesized contents yields
%e A303950 (...), (...), 1, (...), 2, 4, (...), 9, 1, 3, (....), 5, 2, 4, 6, (.....), 7, 9, 1, 3, 8, (.....), 13, 5, 2, 4, 6, 10, (.....), 11, ...
%e A303950 We see that the remaining terms slowly rebuild the starting sequence.
%Y A303950 Cf. A000045 (Fibonacci numbers).
%Y A303950 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 A303950 nonn,base
%O A303950 1,2
%A A303950 _Eric Angelini_ and _Lars Blomberg_, May 03 2018