OFFSET
0,3
COMMENTS
The cost of a request equals the position of the requested element in the list.
After a request, the requested element is moved to the front of the list.
REFERENCES
D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, 1998, pp. 401-403.
LINKS
Ran Bachrach and Ran El-Yaniv, Online list accessing algorithms and their applications: recent empirical evidence, Proceedings of the 8th annual ACM-SIAM symposium on discrete algorithms, SODA ’97, New Orleans, LA, January 5-7, 1997, 53-62.
Wikipedia, Move-to-front transform.
Wikipedia, Self-organizing list.
FORMULA
The sum of a(j) over all j such that A000120(j) = k (number of requests) and A333766(j) <= m (upper bound on the requested elements) equals m^k * k * (m+1)/2. This is a consequence of the fact that the first m positions of the list are occupied by the elements 1, ..., m, as long as no element larger than m has been requested so far.
EXAMPLE
For n=931 (the smallest n for which A374993(n), A374994(n), A374995(n), and a(n) are all distinct), the 931st composition is (1, 1, 2, 4, 1, 1), giving the following development of the list:
list | position of requested element
--------+------------------------------
1 2 3 4 | 1
^ |
1 2 3 4 | 1
^ |
1 2 3 4 | 2
^ |
2 1 3 4 | 4
^ |
4 2 1 3 | 3
^ |
1 4 2 3 | 1
^ |
---------------------------------------
a(931) = 12
CROSSREFS
KEYWORD
nonn
AUTHOR
Pontus von Brömssen, Jul 27 2024
STATUS
approved