This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A374992 #8 Jul 28 2024 17:00:31 %S A374992 0,1,2,2,3,4,3,3,4,5,3,5,4,5,4,4,5,6,6,6,5,5,6,6,5,6,4,6,5,6,5,5,6,7, %T A374992 7,7,4,9,8,7,6,8,4,6,7,8,7,7,6,7,7,7,6,6,7,7,6,7,5,7,6,7,6,6,7,8,8,8, %U A374992 8,10,9,8,7,6,7,10,7,10,9,8,7,9,7,9,6,6 %N A374992 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 move-to-front updating strategy. %C A374992 The cost of a request equals the position of the requested element in the list. %C A374992 After a request, the requested element is moved to the front of the list. %D A374992 D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, 1998, pp. 401-403. %H A374992 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 A374992 Wikipedia, <a href="https://en.wikipedia.org/wiki/Move-to-front_transform">Move-to-front transform</a>. %H A374992 Wikipedia, <a href="https://en.wikipedia.org/wiki/Self-organizing_list">Self-organizing list</a>. %F A374992 a(n) = A374996(k,n) whenever k >= A333766(n)-1. %F A374992 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 A374992 a(n) = a(A025480(n-1)) + A374997(n) for n >= 1. %e A374992 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: %e A374992 list | position of requested element %e A374992 --------+------------------------------ %e A374992 1 2 3 4 | 1 %e A374992 ^ | %e A374992 1 2 3 4 | 1 %e A374992 ^ | %e A374992 1 2 3 4 | 2 %e A374992 ^ | %e A374992 2 1 3 4 | 4 %e A374992 ^ | %e A374992 4 2 1 3 | 3 %e A374992 ^ | %e A374992 1 4 2 3 | 1 %e A374992 ^ | %e A374992 --------------------------------------- %e A374992 a(931) = 12 %Y A374992 Analogous sequences for other updating strategies: A374993, A374994, A374995, A374996. %Y A374992 Cf. A000120, A025480, A066099 (compositions in standard order), A333766, A374997. %K A374992 nonn %O A374992 0,3 %A A374992 _Pontus von Brömssen_, Jul 27 2024