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A303951 A fractal-like sequence: erasing all pairs of contiguous terms that don't sum up to a Fibonacci number leaves the sequence unchanged.

%I A303951 #11 Dec 02 2019 04:03:56
%S A303951 1,2,3,4,9,5,3,10,6,7,8,13,11,23,12,22,14,20,15,19,16,18,17,4,9,25,21,
%T A303951 34,24,31,26,29,27,28,30,59,32,57,33,56,35,54,36,53,37,52,38,51,39,50,
%U A303951 40,49,41,48,42,47,43,46,44,45,55,89,58,86,60,84,61,83
%N A303951 A fractal-like sequence: erasing all pairs of contiguous terms that don't sum up to a Fibonacci number leaves the sequence unchanged.
%C A303951 The sequence is fractal-like as it embeds an infinite number of copies of itself.
%C A303951 The sequence was built according to these rules (see, in the Example section, the parenthesization technique):
%C A303951   1) no overlapping pairs of parentheses;
%C A303951   2) always start the content inside a pair of parentheses with the smallest integer C > 2 not yet present inside another pair of parentheses;
%C A303951   3) always end the content inside a pair of parentheses with the smallest integer I > 2 not yet present inside another pair of parentheses such that the sum C + I is not a Fibonacci number;
%C A303951   4) after a(1) = 1 and 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 A303951 Lars Blomberg, <a href="/A303951/b303951.txt">Table of n, a(n) for n = 1..999</a>
%e A303951 Parentheses are added around each pair of terms that don't sum up to a Fibonacci:
%e A303951 1, 2, (3,4), (9,5), 3, (10,6), (7,8), (13,11), (23,12), (22,14), (20,15), (19,16), (18,17), 4, 9, (25,21), ...
%e A303951 Erasing all the parenthesized contents yields
%e A303951 1, 2, (...), (...), 3, (....), (...), (.....), (.....), (.....), (.....), (.....), (.....), 4, 9, (.....), ...
%e A303951 We see that the remaining terms slowly rebuild the starting sequence.
%Y A303951 Cf. A000045 (Fibonacci numbers).
%Y A303951 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), A303950 (pair summing up to a Fibonacci).
%K A303951 nonn,base
%O A303951 1,2
%A A303951 _Lars Blomberg_ and _Eric Angelini_, May 03 2018