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 A338721 #29 Jul 09 2025 04:54:34 %S A338721 1,3,5,1,7,1,9,3,11,3,1,13,5,1,15,5,1,17,7,3,19,7,3,1,21,9,3,1,23,9,5, %T A338721 1,25,11,5,1,27,11,5,3,29,13,7,3,1,31,13,7,3,1,33,15,7,3,1,35,15,9,5, %U A338721 1,37,17,9,5,1,39,17,9,5,3,41,19,11,5,3,1,43,19,11,7,3,1 %N A338721 Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists the odd numbers k times, and the first element of column k is in row k(k+1)/2. %C A338721 A missing companion to A196020 and A235791. %C A338721 T(n,k) is the total number of horizontal steps in the first n levels of the k-th largest double-staircase of the diagram defined in A335616 (see example). - _Omar E. Pol_, Nov 30 2020 %C A338721 Column k is the partial sums of the k-th column of A339275. - _Omar E. Pol_, Dec 01 2020 %H A338721 Alois P. Heinz, <a href="/A338721/b338721.txt">Rows n = 1..500, flattened</a> %F A338721 T(n,k) = 2 * floor((n-k*(k-1)/2)/k) - 1. - _Alois P. Heinz_, Nov 30 2020 %e A338721 Triangle begins: %e A338721 1; %e A338721 3; %e A338721 5, 1; %e A338721 7, 1; %e A338721 9, 3; %e A338721 11, 3, 1; %e A338721 13, 5, 1; %e A338721 15, 5, 1; %e A338721 17, 7, 3; %e A338721 19, 7, 3, 1; %e A338721 21, 9, 3, 1; %e A338721 23, 9, 5, 1; %e A338721 25, 11, 5, 1; %e A338721 27, 11, 5, 3; %e A338721 29, 13, 7, 3, 1; %e A338721 31, 13, 7, 3, 1; %e A338721 33, 15, 7, 3, 1; %e A338721 35, 15, 9, 5, 1; %e A338721 37, 17, 9, 5, 1; %e A338721 39, 17, 9, 5, 3; %e A338721 41, 19, 11, 5, 3, 1; %e A338721 43, 19, 11, 7, 3, 1; %e A338721 45, 21, 11, 7, 3, 1; %e A338721 47, 21, 13, 7, 3, 1; %e A338721 49, 23, 13, 7, 5, 1; %e A338721 51, 23, 13, 9, 5, 1; %e A338721 53, 25, 15, 9, 5, 3; %e A338721 55, 25, 15, 9, 5, 3, 1; %e A338721 ... %e A338721 From _Omar E. Pol_, Nov 30 2020: (Start) %e A338721 For an illustration of the rows of triangle consider the infinite "double-staircases" diagram defined in A335616. %e A338721 For n = 15 the diagram with first 15 levels looks like this: %e A338721 . %e A338721 Level "Double-staircases" diagram %e A338721 . _ %e A338721 1 _|1|_ %e A338721 2 _|1 _ 1|_ %e A338721 3 _|1 |1| 1|_ %e A338721 4 _|1 _| |_ 1|_ %e A338721 5 _|1 |1 _ 1| 1|_ %e A338721 6 _|1 _| |1| |_ 1|_ %e A338721 7 _|1 |1 | | 1| 1|_ %e A338721 8 _|1 _| _| |_ |_ 1|_ %e A338721 9 _|1 |1 |1 _ 1| 1| 1|_ %e A338721 10 _|1 _| | |1| | |_ 1|_ %e A338721 11 _|1 |1 _| | | |_ 1| 1|_ %e A338721 12 _|1 _| |1 | | 1| |_ 1|_ %e A338721 13 _|1 |1 | _| |_ | 1| 1|_ %e A338721 14 _|1 _| _| |1 _ 1| |_ |_ 1|_ %e A338721 15 |1 |1 |1 | |1| | 1| 1| 1| %e A338721 . %e A338721 The first largest double-staircase has 29 horizontal steps, the second double-staircase has 13 steps, the third double-staircase has 7 steps, the fourth double-staircase has 3 steps and the fifth double-staircase has only one step, so the 15th row of triangle is [29, 13, 7, 3, 1]. (End) %p A338721 T:= (n, k)-> 2*iquo(n-k*(k-1)/2, k)-1: %p A338721 seq(seq(T(n,k), k=1..floor((sqrt(1+8*n)-1)/2)), n=1..30); # _Alois P. Heinz_, Nov 30 2020 %Y A338721 Cf. A196020, A235791, A237593, A335616, A338722, A338723, A339275. %K A338721 nonn,tabf %O A338721 1,2 %A A338721 _N. J. A. Sloane_, Nov 30 2020