cp's OEIS Frontend

A303954 A fractal-like sequence: erasing all pairs of contiguous terms that don't sum up to a square leaves the sequence unchanged.

%I A303954 #12 Dec 02 2019 04:14:42
%S A303954 1,2,7,3,1,8,4,5,6,10,9,16,11,14,12,13,15,21,17,19,18,31,20,29,22,27,
%T A303954 23,2,7,42,24,25,26,38,28,36,30,34,32,49,33,3,1,8,41,35,46,37,44,39,
%U A303954 61,40,60,43,57,45,4,5,59,47,53,48,52,50,71,51,70,54,67
%N A303954 A fractal-like sequence: erasing all pairs of contiguous terms that don't sum up to a square leaves the sequence unchanged.
%C A303954 The sequence is fractal-like as it embeds an infinite number of copies of itself.
%C A303954 The sequence was built according to these rules (see, in the Example section, the parenthesization technique):
%C A303954   1) no overlapping pairs of parentheses;
%C A303954   2) always start the content inside a pair of parentheses with the smallest integer N not yet present inside another pair of parentheses;
%C A303954   3) always end the content inside a pair of parentheses with the smallest integer E not yet present inside another pair of parentheses such that the sum N + E is not a square number;
%C A303954   4) after a(1) = 1 and a(2) = 2, 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 A303954 Lars Blomberg, <a href="/A303954/b303954.txt">Table of n, a(n) for n = 1..1000</a>
%e A303954 Parentheses are added around each pair of terms that don't sum up to a square:
%e A303954 (1,2), (7,3), 1, (8,4), (5,6), (10,9), (16,11), (14,12), (13,15), (21,17), (19,18), (31,20), (29,22), (27,23), 2, 7, (42,24),
%e A303954 Erasing all the parenthesized contents yields
%e A303954 (...), (...), 1, (...), (...), (....), (.....), (.....), (.....), (.....), (.....), (.....), (.....), (.....), 2, 7, (.....),
%e A303954 We see that the remaining terms slowly rebuild the starting sequence.
%Y A303954 Cf. A000290 (Square numbers).
%Y A303954 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), A303953 (pair summing up to a square).
%K A303954 nonn,base
%O A303954 1,2
%A A303954 _Lars Blomberg_ and _Eric Angelini_, May 03 2018