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 A272120 #42 Oct 10 2022 09:39:35 %S A272120 1,1,3,1,3,5,1,3,4,7,1,3,4,7,9,1,3,4,7,6,11,1,3,4,7,6,11,13,1,3,4,7,6, %T A272120 12,8,15,1,3,4,7,6,12,8,15,17,1,3,4,7,6,12,8,15,10,19,1,3,4,7,6,12,8, %U A272120 15,13,19,21,1,3,4,7,6,12,8,15,13,19,12,23,1,3,4,7,6,12,8,15,13,18,12,23,25,1,3,4,7 %N A272120 Square array T(n,k), n>=1, k>=1, read by antidiagonals downwards in which column k lists the alternating row sums of the first k columns of the triangle A196020. %C A272120 Every column of this square array is associated to an isosceles triangle and to a stepped pyramid in the same way as the sequence A196020 is associated to the isosceles triangle of A237593 and to the pyramid described in A245092. Hence there are infinitely many isosceles triangles and infinitely many pyramids that are associated to this sequence. %C A272120 In the Example section appears the triangles and the top views of the pyramids associated to the columns 1 and 2. %C A272120 The sequence A196020 is associated to the isosceles triangle of A237593 as follows: A196020 --> A236104 --> A235791 --> A237591 --> A237593. Then the structure of the pyramid described in A245092 arises after the 90-degree-zig-zag folding of every row of the isosceles triangle of A237593. %C A272120 Note that the first m terms of column k are also the first m terms of A000203, where m = A000217(k) + k = A000217(k+1) - 1 = A000096(k). %H A272120 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr02.jpg">Illustration of first 28 rows of the isosceles triangle associated to the column k when k tends to infinity</a> %H A272120 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr05.jpg">Perspective view of the first 16 levels of the pyramid associated to the column k when k tends to infinity</a> %e A272120 The corner of the square array begins: %e A272120 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1... %e A272120 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3... %e A272120 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4... %e A272120 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7... %e A272120 9, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6... %e A272120 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12... %e A272120 13, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8... %e A272120 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15... %e A272120 17, 10, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13... %e A272120 19, 19, 19, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18... %e A272120 21, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12... %e A272120 23, 23, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28... %e A272120 25, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14... %e A272120 27, 27, 27, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24... %e A272120 29, 16, 23, 23, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24... %e A272120 ... %e A272120 For k = 1 the first two terms of column k are also the first two terms of A000203, i.e., [1, 3]. %e A272120 For k = 2 the first five terms of column k are also the first five terms of A000203, i.e., [1, 3, 4, 7, 6]. %e A272120 For k = 3 the first nine terms of column k are also the first nine terms of A000203, i.e., [1, 3, 4, 7, 6, 12, 8, 15, 13]. %e A272120 More generally, the first A000096(k) terms of column k are also the first A000096(k) terms of A000203. %e A272120 . %e A272120 Illustration of initial terms of the column 1: %e A272120 . %e A272120 . 2D 3D %e A272120 . Isosceles triangle Top view of the pyramid %e A272120 . before folding after folding %e A272120 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ %e A272120 n _|_ T(n,1) _ _ _ _ _ _ _ _ _ x %e A272120 1 _|_|_|_ 1 |_| | | | | | | | %e A272120 2 y _|_ _|_ _|_ x 3 |_ _| | | | | | | %e A272120 3 _|_ _ _|_ _ _|_ 5 |_ _ _| | | | | | %e A272120 4 _|_ _ _ _|_ _ _ _|_ 7 |_ _ _ _| | | | | %e A272120 5 _|_ _ _ _ _|_ _ _ _ _|_ 9 |_ _ _ _ _| | | | %e A272120 6 _|_ _ _ _ _ _|_ _ _ _ _ _|_ 11 |_ _ _ _ _ _| | | %e A272120 7 _|_ _ _ _ _ _ _|_ _ _ _ _ _ _|_ 13 |_ _ _ _ _ _ _| | %e A272120 8 |_ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _| 15 |_ _ _ _ _ _ _ _| %e A272120 . | %e A272120 . y %e A272120 . %e A272120 Illustration of initial terms of the column 2: %e A272120 . %e A272120 . 2D 3D %e A272120 . Isosceles triangle Top view of the pyramid %e A272120 . before folding after folding %e A272120 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ %e A272120 n _|_ T(n,2) _ _ _ _ _ _ _ _ _ x %e A272120 1 _|_|_|_ 1 |_| | | | | | | | %e A272120 2 y _|_ _|_ _|_ x 3 |_ _|_| | | | | | %e A272120 3 _|_ _|_|_|_ _|_ 4 |_ _| _|_| | | | %e A272120 4 _|_ _ _|_|_|_ _ _|_ 7 |_ _ _| _ _|_| | %e A272120 5 _|_ _ _|_ _|_ _|_ _ _|_ 6 |_ _ _| | _ _ _| %e A272120 6 _|_ _ _ _|_ _|_ _|_ _ _ _|_ 11 |_ _ _ _| | %e A272120 7 _|_ _ _ _|_ _ _|_ _ _|_ _ _ _|_ 8 |_ _ _ _| | %e A272120 8 |_ _ _ _ _|_ _ _|_ _ _|_ _ _ _ _| 15 |_ _ _ _ _| %e A272120 . | %e A272120 . y %e A272120 . %e A272120 Illustration of initial terms of the column 3: %e A272120 . %e A272120 . 2D 3D %e A272120 . Isosceles triangle Top view of the pyramid %e A272120 . before folding after folding %e A272120 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ %e A272120 n _|_ T(n,3) _ _ _ _ _ _ _ _ _ x %e A272120 1 _|_|_|_ 1 |_| | | | | | | | %e A272120 2 y _|_ _|_ _|_ x 3 |_ _|_| | | | | | %e A272120 3 _|_ _|_|_|_ _|_ 4 |_ _| _|_| | | | %e A272120 4 _|_ _ _|_|_|_ _ _|_ 7 |_ _ _| _|_| | %e A272120 5 _|_ _ _|_ _|_ _|_ _ _|_ 6 |_ _ _| _| _ _| %e A272120 6 _|_ _ _ _|_|_|_|_|_ _ _ _|_ 12 |_ _ _ _| _| %e A272120 7 _|_ _ _ _|_ _|_|_|_ _|_ _ _ _|_ 8 |_ _ _ _| | %e A272120 8 |_ _ _ _ _|_ _|_|_|_ _|_ _ _ _ _| 15 |_ _ _ _ _| %e A272120 . | %e A272120 . y %e A272120 . %Y A272120 Column 1 is A005408. %Y A272120 Every diagonal starting with 1 gives A000203. %Y A272120 Columns converge to A000203. %Y A272120 Compare A245093. %Y A272120 Cf. A000096, A000217, A000290, A000330, A024916, A175254, A196020, A235791, A236104, A237591, A237593, A237270, A237271, A244050, A245092, A262626. %K A272120 nonn,tabl %O A272120 1,3 %A A272120 _Omar E. Pol_, Apr 20 2016