%I #9 Dec 02 2019 04:14:38
%S 1,2,3,4,5,6,10,4,7,9,5,6,8,17,10,4,7,11,14,9,5,6,8,12,13,17,10,4,7,
%T 11,15,21,14,9,5,6,8,12,16,20,13,17,10,4,7,11,15,18,31,21,14,9,5,6,8,
%U 12,16,19,30,20,13,17,10,4,7,11,15,18,22,27,31,21,14
%N A fractal-like sequence: erasing all pairs of contiguous terms that sum up to a square 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 R > 3 not yet present inside another pair of parentheses;
%C 3) always end the content inside a pair of parentheses with the smallest integer E > 3 not yet present inside another pair of parentheses such that the sum R + E is a square number;
%C 4) after a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 4 and a(5) = 5, always try to extend the sequence with a duplicate of the oldest term > 5 of the sequence not yet duplicated; if this leads to a contradiction, open a new pair of parentheses.
%H Lars Blomberg, <a href="/A303953/b303953.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 square:
%e 1, 2, 3, (4,5), (6,10), 4, (7,9), 5, 6, (8,17), 10, 4, 7, (11,14), 9, 5, 6, 8, (12,13), 17, 10,
%e Erasing all the parenthesized contents yields
%e 1, 2, 3, (...), (....), 4, (...), 5, 6, (....), 10, 4, 7, (.....), 9, 5, 6, 8, (.....), 17, 10,
%e We see that the remaining terms slowly rebuild the starting sequence.
%Y Cf. A000290 (Square 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), A303950 (pair summing up to a Fibonacci), A303951 (pair not summing up to a Fibonacci).
%K nonn,base
%O 1,2
%A _Lars Blomberg_ and _Eric Angelini_, May 03 2018