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 A347263 #127 Feb 27 2022 22:19:10 %S A347263 1,4,6,16,12,36,1,20,0,64,0,30,6,90,0,42,0,144,17,56,0,156,0,72,34,1, %T A347263 256,0,0,90,0,0,324,10,0,110,0,0,400,0,8,132,70,0,342,0,0,156,0,0,576, %U A347263 121,0,182,0,25,462,0,0,210,102,0,784,0,0,1,240,0,0,0,900,24,52,0,272,0,0,0 %N A347263 Irregular triangle read by rows: T(n,k) is the sum of the subparts of the ziggurat diagram of n (described in A347186) that arise from the (2*k-1)-th double-staircase of the double-staircases diagram of n (described in A335616), n >= 1, k >= 1, and the first element of column k is in row A000384(k). %C A347263 Conjecture 1: the number of nonzero terms in row n equals A082647(n). %C A347263 Conjecture 2: column k lists positive integers interleaved with 2*k+2 zeros. %C A347263 The subparts of the ziggurat diagram are the polygons formed by the cells that are under the staircases. %C A347263 The connection of the subparts of the ziggurat diagram with the polygonal numbers is as follows: %C A347263 The area under a double-staircase labeled with the number j is equal to the m-th (j+2)-gonal number plus the (m-1)-th (j+2)-gonal number, where m is the number of steps on one side of the ladder from the base to the top. %C A347263 The area under a simple-staircase labeled with the number j is equal to the m-th (j+2)-gonal number, where m is the number of steps. %C A347263 So the k-th column of the triangle is related to the (2*k+1)-gonal numbers, for example: %C A347263 For the calculation of column 1 we use triangular numbers A000217. %C A347263 For the calculation of column 2 we use pentagonal numbers A000326. %C A347263 For the calculation of column 3 we use heptagonal numbers A000566. %C A347263 For the calculation of column 4 we use enneagonal numbers A001106. %C A347263 And so on. %C A347263 More generally, for the calculation of column k we use the (2*k+1)-gonal numbers. %C A347263 For further information about the ziggurat diagram see A347186. %e A347263 Triangle begins: %e A347263 n / k 1 2 3 4 %e A347263 ------------------------------ %e A347263 1 | 1; %e A347263 2 | 4; %e A347263 3 | 6; %e A347263 4 | 16; %e A347263 5 | 12; %e A347263 6 | 36, 1; %e A347263 7 | 20, 0; %e A347263 8 | 64, 0; %e A347263 9 | 30, 6; %e A347263 10 | 90, 0; %e A347263 11 | 42, 0; %e A347263 12 | 144, 17; %e A347263 13 | 56, 0; %e A347263 14 | 156, 0; %e A347263 15 | 72, 34, 1; %e A347263 16 | 256, 0, 0; %e A347263 17 | 90, 0, 0; %e A347263 18 | 324, 10, 0; %e A347263 19 | 110, 0 0; %e A347263 20 | 400, 0, 8; %e A347263 21 | 132, 70, 0; %e A347263 22 | 342, 0, 0; %e A347263 23 | 156, 0, 0; %e A347263 24 | 576, 121, 0; %e A347263 25 | 182, 0, 25; %e A347263 26 | 462, 0, 0; %e A347263 27 | 210, 102, 0; %e A347263 28 | 784, 0, 0, 1; %e A347263 ... %e A347263 For n = 15 the calculation of the 15th row of the triangle (in accordance with the geometric algorithm described in A347186) is as follows: %e A347263 Stage 1 (Construction): %e A347263 We draw the diagram called "double-staircases" with 15 levels described in A335616. %e A347263 Then we label the five double-staircases (j = 1..5) as shown below: %e A347263 _ %e A347263 _| |_ %e A347263 _| _ |_ %e A347263 _| | | |_ %e A347263 _| _| |_ |_ %e A347263 _| | _ | |_ %e A347263 _| _| | | |_ |_ %e A347263 _| | | | | |_ %e A347263 _| _| _| |_ |_ |_ %e A347263 _| | | _ | | |_ %e A347263 _| _| | | | | |_ |_ %e A347263 _| | _| | | |_ | |_ %e A347263 _| _| | | | | |_ |_ %e A347263 _| | | _| |_ | | |_ %e A347263 _| _| _| | _ | |_ |_ |_ %e A347263 |_ _ _ _ _ _ _ _|_ _ _|_ _|_|_|_|_ _|_ _ _|_ _ _ _ _ _ _ _| %e A347263 1 2 3 4 5 %e A347263 . %e A347263 Stage 2 (Debugging): %e A347263 We remove the fourth double-staircase as it does not have at least one step at level 1 of the diagram starting from the base, as shown below: %e A347263 _ %e A347263 _| |_ %e A347263 _| _ |_ %e A347263 _| | | |_ %e A347263 _| _| |_ |_ %e A347263 _| | _ | |_ %e A347263 _| _| | | |_ |_ %e A347263 _| | | | | |_ %e A347263 _| _| _| |_ |_ |_ %e A347263 _| | | | | |_ %e A347263 _| _| | | |_ |_ %e A347263 _| | _| |_ | |_ %e A347263 _| _| | | |_ |_ %e A347263 _| | | | | |_ %e A347263 _| _| _| _ |_ |_ |_ %e A347263 |_ _ _ _ _ _ _ _|_ _ _|_ _ _|_|_ _ _|_ _ _|_ _ _ _ _ _ _ _| %e A347263 1 2 3 5 %e A347263 . %e A347263 Stage 3 (Annihilation): %e A347263 We delete the second double-staircase and the steps of the first double-staircase that are just above the second double-staircase. %e A347263 The new diagram has two double-staircases and two simple-staircases as shown below: %e A347263 _ %e A347263 | | %e A347263 _ | | _ %e A347263 _| | _| |_ | |_ %e A347263 _| | | | | |_ %e A347263 _| | | | | |_ %e A347263 _| | _| |_ | |_ %e A347263 _| | | | | |_ %e A347263 _| | | | | |_ %e A347263 _| | _| _ |_ | |_ %e A347263 |_ _ _ _ _ _ _ _|_ _ _|_ _ _|_|_ _ _|_ _ _|_ _ _ _ _ _ _ _| %e A347263 1 3 5 %e A347263 . %e A347263 The diagram is called "ziggurat of 15". %e A347263 Now we calculate the area (or the number of cells) under the staircases with multiplicity using polygonal numbers as shown below: %e A347263 The area under the staircase labeled 1 is equal to A000217(8) = 36. There is a pair of these staircases, so T(15,1) = 2*36 = 72. %e A347263 The area under the double-staircase labeled 3 is equal to A000326(4) + A000326(3) = 22 + 12 = 34, so T(15,2) = 34. %e A347263 The area under the double-staircase labeled 5 is equal to A000566(1) + A000566(0) = 1 + 0 = 1, so T(15,3) = 1. %e A347263 Therefore the 15th row of the triangle is [72, 34, 1]. %Y A347263 Row sums give A347186. %Y A347263 Row n has length A351846(n). %Y A347263 Cf. A347529 (analog for the symmetric representation of sigma). %Y A347263 Cf. A000217, A000326, A000384, A000566, A001106, A082647, A139600, A139601, A237270, A237271, A237593, A279387, A279388, A279391, A335616, A346875. %K A347263 nonn,tabf %O A347263 1,2 %A A347263 _Omar E. Pol_, Sep 05 2021