A304232 A fractal-like sequence: erasing all pairs of consecutive terms a(n) and a(n+1) having the property that the last digit of a(n) is the same as the first digit of a(n+1) leaves the sequence unchanged.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 11, 21, 13, 12, 11, 21, 22, 20, 13, 12, 11, 21, 22, 14, 40, 20, 13, 12, 11, 21, 22, 14, 15, 50, 40, 20, 13, 12, 11, 21, 22, 14, 15, 16, 60, 50, 40, 20, 13, 12, 11, 21, 22, 14, 15, 16, 17, 70, 60, 50, 40, 20, 13, 12, 11, 21, 22, 14, 15, 16, 17, 18, 80

Offset: 1

Author

Eric Angelini, May 08 2018

Keywords

• nonn
• base

Comments

The sequence is fractal-like as it contains 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 S > 10 not yet present inside another pair of parentheses;

3) always end the content inside a pair of parentheses with the smallest integer T > 10 not yet present inside another pair of parentheses such that the integer S ends with a digit d and the integer T starts with the same digit d;

4) after a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 4, a(5) = 5, a(6) = 6, a(7) = 7, a(8) = 8, a(9) = 9, a(10) = 10, always try to extend the sequence with a duplicate > 10 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 such that the last digit of a(n) is the same as the first digit of a(n+1):
1,2,3,4,5,6,7,8,9,10,(11,12),11,(21,13),12,11,21,(22,20),13,12,11,21,22,(14,40),20,13,12,11,21,22,14,(15,50),40,20,
Erasing all the parenthesized contents yields
1,2,3,4,5,6,7,8,9,10,(.....),11,(.....),12,11,21,(.....),13,12,11,21,22,(.....),20,13,12,11,21,22,14,(.....),40,20,
We see that the remaining terms slowly rebuild the starting sequence.
		

Links

• Eric Angelini, Table of n, a(n) for n = 1..585

Crossrefs

Cf. A303845 or A303948 (where the erasure techniques are different).