A303951 - OEIS

%I #11 Dec 02 2019 04:03:56

%S 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 34,24,31,26,29,27,28,30,59,32,57,33,56,35,54,36,53,37,52,38,51,39,50,

%U 40,49,41,48,42,47,43,46,44,45,55,89,58,86,60,84,61,83

%N A fractal-like sequence: erasing all pairs of contiguous terms that don't 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 C > 2 not yet present inside another pair of parentheses;

%C   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   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 Lars Blomberg, <a href="/A303951/b303951.txt">Table of n, a(n) for n = 1..999</a>

%e Parentheses are added around each pair of terms that don't sum up to a Fibonacci:

%e 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 Erasing all the parenthesized contents yields

%e 1, 2, (...), (...), 3, (....), (...), (.....), (.....), (.....), (.....), (.....), (.....), 4, 9, (.....), ...

%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), A303950 (pair summing up to a Fibonacci).

%K nonn,base

%O 1,2

%A _Lars Blomberg_ and _Eric Angelini_, May 03 2018