A374994 - OEIS

%I #5 Jul 28 2024 17:00:53

%S 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,

%T 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,

%U 8,10,9,8,7,6,7,10,7,10,9,8,7,9,7,9,6,6

%N 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.

%C The cost of a request equals the position of the requested element in the list.

%C 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.

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

%H Ran Bachrach and Ran El-Yaniv, <a href="https://dl.acm.org/doi/10.5555/314161.314180">Online list accessing algorithms and their applications: recent empirical evidence</a>, Proceedings of the 8th annual ACM-SIAM symposium on discrete algorithms, SODA ’97, New Orleans, LA, January 5-7, 1997, 53-62.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Self-organizing_list">Self-organizing list</a>.

%F 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.

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

%e 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:

%e    list   | position of requested element

%e   --------+------------------------------

%e   1 2 3 4 |         1

%e   ^       |

%e   1 2 3 4 |         1

%e   ^       |

%e   1 2 3 4 |         2

%e     ^     |

%e   1 2 3 4 |         4

%e         ^ |

%e   1 4 2 3 |         1

%e   ^       |

%e   1 4 2 3 |         1

%e   ^       |

%e   ---------------------------------------

%e           a(931) = 10

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

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

%K nonn

%O 0,3

%A _Pontus von Brömssen_, Jul 27 2024