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A303953 A fractal-like sequence: erasing all pairs of contiguous terms that sum up to a square leaves the sequence unchanged. 4
1, 2, 3, 4, 5, 6, 10, 4, 7, 9, 5, 6, 8, 17, 10, 4, 7, 11, 14, 9, 5, 6, 8, 12, 13, 17, 10, 4, 7, 11, 15, 21, 14, 9, 5, 6, 8, 12, 16, 20, 13, 17, 10, 4, 7, 11, 15, 18, 31, 21, 14, 9, 5, 6, 8, 12, 16, 19, 30, 20, 13, 17, 10, 4, 7, 11, 15, 18, 22, 27, 31, 21, 14 (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 R > 3 not yet present inside another pair of parentheses;

  3) always end the content inside a pair of parentheses with the smallest integer E > 3 not yet present inside another pair of parentheses such that the sum R + E is a square number;

  4) after a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 4 and a(5) = 5, 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.

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 square:

1, 2, 3, (4,5), (6,10), 4, (7,9), 5, 6, (8,17), 10, 4, 7, (11,14), 9, 5, 6, 8, (12,13), 17, 10,

Erasing all the parenthesized contents yields

1, 2, 3, (...), (....), 4, (...), 5, 6, (....), 10, 4, 7, (.....), 9, 5, 6, 8, (.....), 17, 10,

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

CROSSREFS

Cf. A000290 (Square 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), A303951 (pair not summing up to a Fibonacci).

Sequence in context: A076299 A136683 A200332 * A195185 A194843 A194913

Adjacent sequences:  A303950 A303951 A303952 * A303954 A303955 A303956

KEYWORD

nonn,base

AUTHOR

Lars Blomberg and Eric Angelini, May 03 2018

STATUS

approved

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Last modified September 23 03:35 EDT 2020. Contains 337291 sequences. (Running on oeis4.)