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 A346875 #39 Dec 10 2021 11:19:35 %S A346875 1,4,1,1,8,3,2,1,1,15,5,3,2,1,1,1,23,8,5,2,2,2,1,1,1,34,11,6,4,3,2,2, %T A346875 1,1,1,1,46,16,8,5,4,2,3,1,2,1,1,1,1,61,20,11,6,5,3,3,2,2,2,1,1,1,1,1, %U A346875 77,26,14,8,5,5,3,2,3,2,1,2,1,1,1,1,1,96,32,16 %N A346875 Irregular triangle read by rows in which row n lists the row A000384(n) of A237591, n >= 1. %C A346875 The characteristic shape of the symmetric representation of sigma(A000384(n)) consists in that in the main diagonal of the diagram the smallest Dyck path has a valley and the largest Dyck path has a peak. %C A346875 So knowing this we can know if a number is a hexagonal number (or not) just by looking at the diagram, even ignoring the concept of hexagonal number. %C A346875 Therefore we can see a geometric pattern of the distribution of the hexagonal numbers in the stepped pyramid described in A245092. %C A346875 T(n,k) is also the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(A000384(n-1)), from the border to the center, hence the sum of the n-th row of triangle is equal to A000384(n-1). %C A346875 T(n,k) is also the difference between the total number of partitions of all positive integers <= n-th hexagonal number into exactly k consecutive parts, and the total number of partitions of all positive integers <= n-th hexagonal number into exactly k + 1 consecutive parts. %e A346875 Triangle begins: %e A346875 1; %e A346875 4, 1, 1; %e A346875 8, 3, 2, 1, 1; %e A346875 15, 5, 3, 2, 1, 1, 1; %e A346875 23, 8, 5, 2, 2, 2, 1, 1, 1; %e A346875 34, 11, 6, 4, 3, 2, 2, 1, 1, 1, 1; %e A346875 46, 16, 8, 5, 4, 2, 3, 1, 2, 1, 1, 1, 1; %e A346875 61, 20, 11, 6, 5, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1; %e A346875 77, 26, 14, 8, 5, 5, 3, 2, 3, 2, 1, 2, 1, 1, 1, 1, 1; %e A346875 96, 32, 16, 10, 7, 5, 4, 3, 3, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1; %e A346875 ... %e A346875 Illustration of initial terms: %e A346875 Column H gives the nonzero hexagonal numbers (A000384). %e A346875 Column S gives the sum of the divisors of the hexagonal numbers which equals the area (and the number of cells) of the associated diagram. %e A346875 ------------------------------------------------------------------------- %e A346875 n H S Diagram %e A346875 ------------------------------------------------------------------------- %e A346875 _ _ _ _ %e A346875 1 1 1 |_| | | | | | | %e A346875 1 | | | | | | %e A346875 _ _| | | | | | %e A346875 | _| | | | | %e A346875 _ _ _| _| | | | | %e A346875 2 6 12 |_ _ _ _| 1 | | | | %e A346875 4 1 | | | | %e A346875 _ _ _|_| | | %e A346875 _ _| | | | %e A346875 | _| | | %e A346875 _| _| | | %e A346875 |_ _|1 1 | | %e A346875 | 2 | | %e A346875 _ _ _ _ _ _ _ _|4 _ _ _ _ _| | %e A346875 3 15 24 |_ _ _ _ _ _ _ _| | _ _ _ _ _| %e A346875 8 | | %e A346875 _ _| | %e A346875 _ _| _ _| %e A346875 | _| %e A346875 _| _| %e A346875 | _|1 1 %e A346875 _ _ _| | 1 %e A346875 | _ _ _|2 %e A346875 | | 3 %e A346875 | | %e A346875 | |5 %e A346875 _ _ _ _ _ _ _ _ _ _ _ _ _ _| | %e A346875 4 28 56 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| %e A346875 15 %e A346875 . %Y A346875 Row sums give A000384, n >= 1. %Y A346875 Row lengths give A005408. %Y A346875 Column 1 is A267682, n >= 1. %Y A346875 For the characteristic shape of sigma(A000040(n)) see A346871. %Y A346875 For the characteristic shape of sigma(A000079(n)) see A346872. %Y A346875 For the characteristic shape of sigma(A000217(n)) see A346873. %Y A346875 For the visualization of Mersenne numbers A000225 see A346874. %Y A346875 For the characteristic shape of sigma(A000396(n)) see A346876. %Y A346875 For the characteristic shape of sigma(A008588(n)) see A224613. %Y A346875 Cf. A000203, A237591, A237593, A245092, A249351, A262626. %K A346875 nonn,tabf %O A346875 1,2 %A A346875 _Omar E. Pol_, Aug 06 2021