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 A383330 #7 Apr 26 2025 11:27:27 %S A383330 0,0,1,1,2,2,2,3,3,3,2,3,3,4,3,3,4,4,4,4,4,3,4,4,4,4,5,4,4,5,5,5,5,5, %T A383330 5,5,3,4,4,5,4,5,5,6,4,4,5,5,5,5,5,5,6,5,5,4,5,5,5,5,5,5,6,5,6,5,5,6, %U A383330 6,6,6,6,6,6,6,6,6,6,4,5,5,5,5,6,5,6,5,6,6,7,5 %N A383330 Triangle read by rows: T(n,k) is the length of a shortest vectorial addition chain for (n,k), 0 <= k <= n. %C A383330 Starting with (1,0) and (0,1), each pair of the chain must be equal to the sum of two preceding pairs. The length of the chain is defined to be the number of pairs in the chain, excluding (1,0) and (0,1). %C A383330 Also, T(n,k) is the least number of multiplications needed to obtain x^n*y^k, starting with x and y. %C A383330 T(0,0) = 0 by convention. %H A383330 Richard Bellman, <a href="https://doi.org/10.2307/2312273">Problem 5125</a>, Advanced problems and solutions, The American Mathematical Monthly 70 (1963), p. 765. %H A383330 Jorge Olivos, <a href="https://doi.org/10.1016/0196-6774(81)90003-1">On vectorial addition chains</a>, Journal of Algorithms 2 (1981), 13-21. %H A383330 E. G. Straus, <a href="https://doi.org/10.2307/2310929">Partial solution to problem 5125: Addition chains of vectors</a>, Problems and solutions, The American Mathematical Monthly 71 (1964), 806-808. %H A383330 Edward G. Thurber and Neill M. Clift, <a href="https://doi.org/10.1016/j.disc.2020.112200">Addition chains, vector chains, and efficient computation</a>, Discrete Mathematics 344 (2021), 112200. %H A383330 Wikipedia, <a href="https://en.wikipedia.org/wiki/Vectorial_addition_chain">Vectorial addition chain</a>. %e A383330 Triangle begins: %e A383330 n\k| 0 1 2 3 4 5 6 7 8 9 10 %e A383330 ---+-------------------------------- %e A383330 0 | 0 %e A383330 1 | 0 1 %e A383330 2 | 1 2 2 %e A383330 3 | 2 3 3 3 %e A383330 4 | 2 3 3 4 3 %e A383330 5 | 3 4 4 4 4 4 %e A383330 6 | 3 4 4 4 4 5 4 %e A383330 7 | 4 5 5 5 5 5 5 5 %e A383330 8 | 3 4 4 5 4 5 5 6 4 %e A383330 9 | 4 5 5 5 5 5 5 6 5 5 %e A383330 10 | 4 5 5 5 5 5 5 6 5 6 5 %e A383330 A shortest addition chain for (11,7) is [(1,0), (0,1),] (1,1), (2,1), (4,2), (5,3), (10,6), (11,7) of length T(11,7) = 6. %Y A383330 Cf. A003313 (column k=0, excluding T(0,0)), A265690 (column k=1 and main diagonal; apparently also column k=2), A383331, A383332. %K A383330 nonn,tabl %O A383330 0,5 %A A383330 _Pontus von Brömssen_, Apr 26 2025