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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. 6
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, 7, 9, 1, 3, 8, 12, 15, 19, 22, 13, 5, 2, 4, 6, 10, 14, 16, 18, 20, 11, 7, 9, 1, 3, 8, 12, 15, 17, 38, 19, 22, 13, 5, 2, 4, 6, 10, 14, 16 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The sequence is fractal-like as it embeds an infinite number of copies of itself.

The sequence was built according to these rules (see, in the Example section, the parenthesization technique):

  1) no overlapping pairs of parentheses;

  2) always start the content inside a pair of parentheses with the smallest integer F not yet present inside another pair of parentheses;

  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;

  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.

LINKS

Lars Blomberg, Table of n, a(n) for n = 1..998

EXAMPLE

Parentheses are added around each pair of terms that sum up to a Fibonacci:

(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, ...

Erasing all the parenthesized contents yields

(...), (...), 1, (...), 2, 4, (...), 9, 1, 3, (....), 5, 2, 4, 6, (.....), 7, 9, 1, 3, 8, (.....), 13, 5, 2, 4, 6, 10, (.....), 11, ...

We see that the remaining terms slowly rebuild the starting sequence.

CROSSREFS

Cf. A000045 (Fibonacci numbers), A303845 for another "erasing criteria" (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).

Sequence in context: A062286 A071071 A195729 * A011181 A134822 A229351

Adjacent sequences:  A303947 A303948 A303949 * A303951 A303952 A303953

KEYWORD

nonn,base

AUTHOR

Eric Angelini and Lars Blomberg, May 03 2018

STATUS

approved

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Last modified August 19 08:08 EDT 2019. Contains 326115 sequences. (Running on oeis4.)