A303950 A fractal-like sequence: erasing all pairs of contiguous terms that sum up to a Fibonacci number leaves the sequence unchanged.

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

Offset: 1

Author

Eric Angelini and Lars Blomberg, May 03 2018

Keywords

• nonn
• base

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.

Examples

			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.
		

Links

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

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).