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 A332266 #33 Dec 21 2020 07:29:40 %S A332266 1,2,3,4,1,5,3,6,2,7,4,8,3,9,5,1,10,4,3,11,6,5,12,5,2,13,7,4,14,6,6, %T A332266 15,8,3,16,7,5,1,17,9,7,3,18,8,4,5,19,10,6,7,20,9,8,2,21,11,5,4,22,10, %U A332266 7,6,23,12,9,8,24,11,6,3,25,13,8,5,1,26,12,10,7,3,27,14,7,9,5 %N A332266 Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists successive blocks of k consecutive integers that differ by 2, where the m-th block starts with m, m >= 1, and the first element of column k is in row k^2. %C A332266 This triangle can be interpreted as a table of partitions into consecutive parts that differ by 2 (see the Example section). %e A332266 Triangle begins: %e A332266 1; %e A332266 2; %e A332266 3; %e A332266 4, 1; %e A332266 5, 3; %e A332266 6, 2; %e A332266 7, 4; %e A332266 8, 3; %e A332266 9, 5, 1; %e A332266 10, 4, 3; %e A332266 11, 6, 5; %e A332266 12, 5, 2; %e A332266 13, 7, 4; %e A332266 14, 6, 6; %e A332266 15, 8, 3; %e A332266 16, 7, 5, 1; %e A332266 17, 9, 7, 3; %e A332266 18, 8, 4, 5; %e A332266 19, 10, 6, 7; %e A332266 20, 9, 8, 2; %e A332266 21, 11, 5, 4; %e A332266 22, 10, 7, 6; %e A332266 23, 12, 9, 8; %e A332266 24, 11, 6, 3; %e A332266 25, 13, 8, 5, 1; %e A332266 ... %e A332266 Figures A..G show the location (in the columns of the table) of the partitions of n = 1..7 (respectively) into consecutive parts that differ by 2: %e A332266 . --------------------------------------------------------- %e A332266 Fig: A B C D E F G %e A332266 . --------------------------------------------------------- %e A332266 . n: 1 2 3 4 5 6 7 %e A332266 Row --------------------------------------------------------- %e A332266 1 | [1];| 1; | 1; | 1; | 1; | 1; | 1; | %e A332266 2 | | [2];| 2; | 2; | 2; | 2; | 2; | %e A332266 3 | | | [3];| 3; | 3; | 3; | 3; | %e A332266 4 | | | | [4],[1];| 4, 1;| 4, 1; | 4, 1;| %e A332266 5 | | | | 5, [3];| [5], 3;| 5, 3; | 5, 3;| %e A332266 6 | | | | | | [6],[2];| 6, 2;| %e A332266 7 | | | | | | 7, [4];| [7], 4;| %e A332266 . --------------------------------------------------------- %e A332266 Figure F: for n = 6 the partitions of 6 into consecutive parts that differ by 2 (but with the parts in increasing order) are [6] and [2, 4]. These partitions have one part and two parts respectively. On the other hand we can find the mentioned partitions in the columns 1 and 2 of this table, starting at the row 6. %e A332266 . %e A332266 Figures H..L show the location (in the columns of the table) of the partitions of 8..12 (respectively) into consecutive parts that differ by 2: %e A332266 . ----------------------------------------------------------------------- %e A332266 Fig: H I J K L %e A332266 . ----------------------------------------------------------------------- %e A332266 . n: 8 9 10 11 12 %e A332266 Row ----------------------------------------------------------------------- %e A332266 1 | 1; | 1; | 1; | 1; | 1; | %e A332266 1 | 2; | 2; | 2; | 2; | 2; | %e A332266 3 | 3; | 3; | 3; | 3; | 3; | %e A332266 4 | 4, 1; | 4, 1; | 4, 1; | 4, 1; | 4, 1; | %e A332266 5 | 5, 3; | 5, 3; | 5, 3; | 5, 3; | 5, 3; | %e A332266 6 | 6, 2; | 6, 2; | 6, 2; | 6, 2; | 6, 2; | %e A332266 7 | 7, 4; | 7, 4; | 7, 4; | 7, 4; | 7, 4; | %e A332266 8 | [8],[3]; | 8, 3; | 8, 3; | 8, 3; | 8, 3; | %e A332266 9 | 9, [5], 1;| [9], 5, [1];| 9, 5, 1;| 9, 5, 1;| 9, 5, 1; | %e A332266 10 | | 10, 4, [3];| [10],[4], 3;| 10, 4, 3;| 10, 4; 3; | %e A332266 11 | | 11, 6, [5];| 11, [6], 5;| [11], 6, 5,| 11, 6; 5; | %e A332266 12 | | | | | [12],[5],[2];| %e A332266 13 | | | | | 13, [7],[4];| %e A332266 14 | | | | | 14, 6, [6];| %e A332266 . ----------------------------------------------------------------------- %e A332266 Figure I: for n = 9 the partitions of 9 into consecutive parts that differ by 2(but with the parts in increasing order) are [9] and [1, 3, 5]. These partitions have one part and three parts respectively. On the other hand, we can find the mentioned partitions in the columns 1 and 3 of this table, starting at the row 9. %e A332266 . %e A332266 Illustration of initial terms arranged into a triangular structure: %e A332266 . _ %e A332266 . _|1| %e A332266 . _|2 | %e A332266 . _|3 _| %e A332266 . _|4 |1| %e A332266 . _|5 _|3| %e A332266 . _|6 |2 | %e A332266 . _|7 _|4 | %e A332266 . _|8 |3 _| %e A332266 . _|9 _|5 |1| %e A332266 . _|10 |4 |3| %e A332266 . _|11 _|6 _|5| %e A332266 . _|12 |5 |2 | %e A332266 . _|13 _|7 |4 | %e A332266 . _|14 |6 _|6 | %e A332266 . _|15 _|8 |3 _| %e A332266 . _|16 |7 |5 |1| %e A332266 . _|17 _|9 _|7 |3| %e A332266 . _|18 |8 |4 |5| %e A332266 . _|19 _|10 |6 _|7| %e A332266 . _|20 |9 _|8 |2 | %e A332266 . _|21 _|11 |5 |4 | %e A332266 . _|22 |10 |7 |6 | %e A332266 . _|23 _|12 _|9 _|8 | %e A332266 . _|24 |11 |6 |3 _| %e A332266 . |25 |13 |8 |5 |1| %e A332266 ... %e A332266 The number of horizontal line segments in the n-th row of the diagram equals A038548(n), the number of partitions of n into consecutive parts that differ by 2. %Y A332266 Tables of the same family where the consecutive parts differ by d are A010766 (d=0), A286001 (d=1), this sequence (d=2), A334945 (d=3), A334618(d=4). %Y A332266 Cf. A038548, A060872, A066839, A303300, A330466. %K A332266 nonn,tabf %O A332266 1,2 %A A332266 _Omar E. Pol_, Feb 08 2020