A374994 Total cost when the elements of the n-th composition (in standard order) are requested from a self-organizing list initialized to (1, 2, 3, ...), using the frequency-count updating strategy.

0, 1, 2, 2, 3, 4, 3, 3, 4, 5, 3, 5, 4, 5, 4, 4, 5, 6, 6, 6, 5, 5, 6, 6, 5, 6, 4, 6, 5, 5, 5, 5, 6, 7, 7, 7, 4, 9, 8, 7, 6, 8, 4, 7, 7, 8, 7, 7, 6, 7, 7, 7, 6, 6, 7, 7, 6, 6, 6, 6, 6, 6, 6, 6, 7, 8, 8, 8, 8, 10, 9, 8, 7, 6, 7, 10, 7, 10, 9, 8, 7, 9, 7, 9, 6, 6

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 so that the list is kept ordered by decreasing number of requests so far. In case of ties, the most recently requested element is placed before all other elements with the same number of requests.

References

D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, 1998, pp. 401-403.

Links

Table of n, a(n) for n=0..85.

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, 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.

a(n) = a(A025480(n-1)) + A374999(n) for n >= 1.

Example

For n=931 (the smallest n for which A374992(n), A374993(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
    ^     |
  1 2 3 4 |         4
        ^ |
  1 4 2 3 |         1
  ^       |
  1 4 2 3 |         1
  ^       |
  ---------------------------------------
          a(931) = 10

Crossrefs

Analogous sequences for other updating strategies: A374992, A374993, A374995, A374996.

Cf. A000120, A025480, A066099 (compositions in standard order), A333766, A374999.

Sequence in context: A374995 A350238 A376937 * A374992 A352746 A330416

Adjacent sequences: A374991 A374992 A374993 * A374995 A374996 A374997

Keyword

nonn

Author

Pontus von Brömssen, Jul 27 2024

Status

approved