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 A384134 #26 May 31 2025 14:39:13 %S A384134 1,1,1,1,2,1,2,6,2,1,1,12,9,2,1,2,23,25,10,2,1,1,45,69,35,10,2,1,5,98, %T A384134 178,119,38,10,2,1,2,278,457,371,151,39,10,2,1 %N A384134 Triangle read by rows: T(n,k) is the number of Cauchy-complete categories with n morphisms and k objects. %C A384134 A Cauchy complete (also called Karoubi complete or idempotent-complete) category is one in which all idempotents split. In other words, in a Cauchy-complete category every arrow e:A->A such that e=e*e has a retract, meaning there exists an object B and morphisms r:A→B and s:B→A such that s∘r=e but r∘s=1_B. %H A384134 Geoff Cruttwell, <a href="https://www.reluctantm.com/gcruttw/publications/ams2014CruttwellCountingFiniteCats.pdf">Counting Finite Categories</a>, presentation, (2018). %H A384134 nLab, <a href="https://ncatlab.org/nlab/show/Cauchy+complete+category">Cauchy complete category</a>. %H A384134 nLab, <a href="https://ncatlab.org/nlab/show/Karoubi+envelope">Karoubi envelope</a>. %F A384134 T(n,k) = A384066(n-k) if k >= (2/3)*n. %F A384134 T(3n,2n) = T(3n-1,2n-1) + 1 when n >= 1. %F A384134 T(3n-1,2n-1) = T(3n-2,2n-2) + 3 when n >= 2. %F A384134 T(3n-2,2n-2) = T(3n-3,2n-3) + 13 when n >= 4. %e A384134 Triangle begins: %e A384134 1; %e A384134 1, 1; %e A384134 1, 2, 1; %e A384134 2, 6, 2, 1; %e A384134 1, 12, 9, 2, 1; %e A384134 2, 23, 25, 10, 2, 1; %e A384134 ... %Y A384134 Cf. A384135 (row sums), A000001 (column 1), A384066 (limiting values), A125697. %K A384134 nonn,tabl,hard,more %O A384134 1,5 %A A384134 _Elijah Beregovsky_, May 20 2025