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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. 4
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, 34, 24, 31, 26, 29, 27, 28, 30, 59, 32, 57, 33, 56, 35, 54, 36, 53, 37, 52, 38, 51, 39, 50, 40, 49, 41, 48, 42, 47, 43, 46, 44, 45, 55, 89, 58, 86, 60, 84, 61, 83 (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 C > 2 not yet present inside another pair of parentheses;

  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;

  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.

LINKS

Lars Blomberg, Table of n, a(n) for n = 1..999

EXAMPLE

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

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), ...

Erasing all the parenthesized contents yields

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

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

CROSSREFS

Cf. A000045 (Fibonacci numbers).

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).

Sequence in context: A124526 A124418 A175177 * A326776 A249746 A112480

Adjacent sequences:  A303948 A303949 A303950 * A303952 A303953 A303954

KEYWORD

nonn,base

AUTHOR

Lars Blomberg and Eric Angelini, May 03 2018

STATUS

approved

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Last modified September 29 06:29 EDT 2020. Contains 337425 sequences. (Running on oeis4.)