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 A384480 #7 Jun 09 2025 10:38:23 %S A384480 0,0,0,1,1,1,2,2,2,2,2,3,2,3,2,3,3,3,3,3,3,3,4,3,3,3,4,3,4,4,4,4,3,4, %T A384480 4,4,3,5,4,4,3,4,4,5,3,4,4,4,4,4,4,4,5,4,3,4,5,4,5,3,4,4,5,4,4,4,5,5, %U A384480 5,5,4,4,4,5,4,5,5,4,4,5,5,5,4,4,4,5,4,4,5,5,4 %N A384480 Square array read by antidiagonals: T(n,k) is the length of a shortest addition-composition chain for n*x+k, starting with 1 and x; n, k >= 0. %C A384480 An addition-composition chain for the affine function f is a finite sequence of affine functions, starting with 1, x and ending with f, in which each element except 1 and x equals g(x)+h(x) or g(h(x)) for two preceding, not necessarily distinct, elements g(x) and h(x) in the chain. The length of the chain is the number of elements in the chain, excluding 1 and x. Such chains exist only for functions of the form f(x) = n*x+k, where n and k are nonnegative integers, not both 0. %C A384480 T(0,0) = 0 by convention. %C A384480 Equivalently, the chains can be defined on pairs (s,t) of nonnegative integers (corresponding to the function f(x) = s*x+t) with the operations (s,t)+(u,v) = (s+t,u+v) (addition) and (s,t)o(u,v) = (s*u,s*v+t) (composition). %F A384480 T(n,k) <= T(n,k-1) + 1. %F A384480 T(n,k) <= T(n-1,k) + 1. %e A384480 Array begins: %e A384480 n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12 %e A384480 ---+-------------------------------------- %e A384480 0 | 0 0 1 2 2 3 3 4 3 4 4 5 4 %e A384480 1 | 0 1 2 3 3 4 4 5 4 5 5 5 5 %e A384480 2 | 1 2 2 3 3 4 4 4 4 5 5 5 5 %e A384480 3 | 2 3 3 3 4 4 4 5 5 5 5 5 5 %e A384480 4 | 2 3 3 3 3 4 3 4 4 4 4 5 4 %e A384480 5 | 3 4 4 4 4 4 4 4 5 5 5 5 5 %e A384480 6 | 3 4 4 4 4 4 4 5 4 4 5 5 5 %e A384480 7 | 4 5 5 5 5 5 5 5 5 5 5 6 6 %e A384480 8 | 3 4 4 4 4 4 4 4 4 5 4 5 4 %e A384480 9 | 3 4 5 4 4 5 5 5 4 5 5 5 4 %e A384480 10 | 4 5 5 5 5 5 5 5 5 5 5 6 5 %e A384480 11 | 4 5 6 5 5 5 6 6 6 5 5 6 6 %e A384480 12 | 4 5 5 5 5 5 5 5 5 5 5 5 5 %e A384480 For (n,k) = (4,6), the unique shortest chain for 4*x+6 is (1, x,) x+1, 2*x+2, 4*x+6 of length T(4,6) = 3. The last term of the chain is the composition of 2*x+2 with itself. %e A384480 For (n,k) = (6,4), a shortest chain for 6*x+4 is (1, x,) x+1, 2*x+2, 3*x+2, 6*x+4 of length T(6,4) = 4. This chain uses only additions. %Y A384480 Cf. A383330 (addition only), A384481, A384482, A384483 (row 0). %K A384480 nonn,tabl %O A384480 0,7 %A A384480 _Pontus von Brömssen_, Jun 02 2025