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 A334945 #26 Dec 19 2020 07:56:27 %S A334945 1,2,3,4,5,1,6,4,7,2,8,5,9,3,10,6,11,4,12,7,1,13,5,4,14,8,7,15,6,2,16, %T A334945 9,5,17,7,8,18,10,3,19,8,6,20,11,9,21,9,4,22,12,7,1,23,10,10,4,24,13, %U A334945 5,7,25,11,8,10,26,14,11,2,27,12,6,5,28,15,9,8,29,13,12,11,30,16,7,3 %N A334945 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 3, where the m-th block starts with m, m >= 1, and the first element of column k is in the row that is the k-th pentagonal number (A000326). %C A334945 This triangle can be interpreted as a table of partitions into consecutive parts that differ by 3 (see the Example section). %C A334945 Also, every triangle of this family has the property that starting from row n the sum of k positive and consecutive terms in the column k is equal to n. - _Omar E. Pol_, Dec 18 2020 %e A334945 Triangle begins: %e A334945 1; %e A334945 2; %e A334945 3; %e A334945 4; %e A334945 5, 1; %e A334945 6, 4; %e A334945 7, 2; %e A334945 8, 5; %e A334945 9, 3; %e A334945 10, 6; %e A334945 11, 4; %e A334945 12, 7, 1; %e A334945 13, 5, 4; %e A334945 14, 8, 7; %e A334945 15, 6, 2; %e A334945 16, 9, 5; %e A334945 17, 7, 8; %e A334945 18, 10, 3; %e A334945 19, 8, 6; %e A334945 20, 11, 9; %e A334945 21, 9, 4; %e A334945 22, 12, 7, 1; %e A334945 ... %e A334945 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 3: %e A334945 . ----------------------------------------------------- %e A334945 Fig: A B C D E F G %e A334945 . ----------------------------------------------------- %e A334945 . n: 1 2 3 4 5 6 7 %e A334945 Row ----------------------------------------------------- %e A334945 1 | [1];| 1; | 1; | 1; | 1; | 1; | 1; | %e A334945 2 | | [2];| 2; | 2; | 2; | 2; | 2; | %e A334945 3 | | | [3];| 3; | 3; | 3; | 3; | %e A334945 4 | | | | [4];| 4; | 4; | 4; | %e A334945 5 | | | | | [5],[1];| 5, 1;| 5, 1; | %e A334945 6 | | | | | 6, [4];| [6],4;| 6, 4; | %e A334945 7 | | | | | | | [7],[2];| %e A334945 8 | | | | | | | 8, [5];| %e A334945 . ----------------------------------------------------- %e A334945 Figure G: for n = 7 the partitions of 7 into consecutive parts that differ by 3 (but with the parts in increasing order) are [7] and [2, 5]. 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 7. %e A334945 . %e A334945 Illustration of initial terms arranged into a triangular structure: %e A334945 . _ %e A334945 . _|1| %e A334945 . _|2 | %e A334945 . _|3 | %e A334945 . _|4 _| %e A334945 . _|5 |1| %e A334945 . _|6 _|4| %e A334945 . _|7 |2 | %e A334945 . _|8 _|5 | %e A334945 . _|9 |3 | %e A334945 . _|10 _|6 | %e A334945 . _|11 |4 _| %e A334945 . _|12 _|7 |1| %e A334945 . _|13 |5 |4| %e A334945 . _|14 _|8 _|7| %e A334945 . _|15 |6 |2 | %e A334945 . _|16 _|9 |5 | %e A334945 . _|17 |7 _|8 | %e A334945 . _|18 _|10 |3 | %e A334945 . _|19 |8 |6 | %e A334945 . _|20 _|11 _|9 | %e A334945 . _|21 |9 |4 _| %e A334945 . |22 |12 |7 |1| %e A334945 ... %e A334945 The number of horizontal line segments in the n-th row of the diagram equals A117277(n), the number of partitions of n into consecutive parts that differ by 3. %Y A334945 Tables of the same family where the consecutive parts differ by d are A010766 (d=0), A286001 (d=1), A332266 (d=2), this sequence (d=3), A334618(d=4). %Y A334945 Cf. A000326, A117277, A330887, A330888, A330889, A334463. %K A334945 nonn,tabf %O A334945 1,2 %A A334945 _Omar E. Pol_, May 27 2020