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 A331499 #49 Mar 31 2020 04:48:05 %S A331499 1,1,1,2,1,2,4,3,2,5,10,6,7,5,14,24,18,15,19,14,42,66,42,48,43,56,42, %T A331499 132,174,130,116,143,132,174,132,429,504,332,374,358,451,423,561,429, %U A331499 1430,1406,1048,974,1182,1163,1475,1397,1859,1430,4862 %N A331499 T(n,k) is the number of nonintersecting arch configurations with n arches on a horizontal axis that have a starting parent arch configuration of k arches. %C A331499 A parent arch configuration has exactly one exterior arch and can be derived from an initial arch configuration with more than one exterior arch by using the algorithm for decreasing arches explained below. %C A331499 When concentric arches are placed below the different arch configurations, the number of components (loops formed) in each configuration will remain the same as the number components in that arch configuration's parent arch. %C A331499 Algorithm: To determine the parent arch configuration, start with all the nonintersecting arch configurations with n arches and more than one exterior arch. For each starting individual arch configuration, reduce the number of arches until there is exactly one exterior arch. Join the start of the first arch with the end of the last arch. This will eliminate one arch and shrink the number of arches from n to n-1. Continue this process until there is one exterior arch. %C A331499 /\ n=5 %C A331499 / \ /\ n=5 %C A331499 / /\\ /\ / \ %C A331499 --> /\//\//\\\ <-- --> //\\//\/\\ <-- %C A331499 /\ %C A331499 --> /\/\//\\ <-- --> /\/\/\/\ <-- %C A331499 /\ /\ %C A331499 --> //\\/\ <-- / \ %C A331499 --> /\/\ <-- //\/\\ parent arch k = 3 %C A331499 /\ parent arch k = 1. %F A331499 The sum of row n = A000108(n). %F A331499 T(n,1) = A000682(n). %F A331499 T(n,n) = A000108(n-1). %F A331499 T(n,n-1) = A000108(n-2). %F A331499 T(n,n-2) = A000108(n-2) + A000108(n-3) (conjectured). %e A331499 Triangle T(n,k) begins: %e A331499 n\k 1 2 3 4 5 6 7 8 9 10 11 %e A331499 1: 1 %e A331499 2: 1 1 %e A331499 3: 2 1 2 %e A331499 4: 4 3 2 5 %e A331499 5: 10 6 7 5 14 %e A331499 6: 24 18 15 19 14 42 %e A331499 7: 66 42 48 43 56 42 132 %e A331499 8: 174 130 116 143 132 174 132 429 %e A331499 9: 504 332 374 358 451 423 561 429 1430 %e A331499 10: 1406 1048 974 1182 1163 1475 1397 1859 1430 4862 %e A331499 11: 4210 2836 3166 3152 3906 3897 4950 4719 6292 4862 16796 %Y A331499 Cf. A000108, A000682. %K A331499 nonn,tabl %O A331499 1,4 %A A331499 _Roger Ford_, Feb 16 2020