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 A346864 #56 Jan 15 2025 08:38:13 %S A346864 2,1,6,2,1,1,11,4,3,1,1,1,19,6,4,2,2,1,1,1,28,10,5,3,3,2,1,1,1,1,40, %T A346864 13,7,5,3,2,2,2,1,1,1,1,53,18,10,5,4,3,3,2,1,2,1,1,1,1,69,23,12,7,5,4, %U A346864 3,2,2,2,2,1,1,1,1,1,86,29,15,9,6,5,4,2,3,2,2,1,2,1,1,1,1,1 %N A346864 Irregular triangle read by rows in which row n lists the row A014105(n) of A237591, n >= 1. %C A346864 The characteristic shape of the symmetric representation of sigma(A014105(n)) consists in that in the main diagonal of the diagram the smallest Dyck path has a peak and the largest Dyck path has a valley. %C A346864 So knowing this characteristic shape we can know if a number is a second hexagonal number (or not) just by looking at the diagram, even ignoring the concept of second hexagonal number. %C A346864 Therefore we can see a geometric pattern of the distribution of the second hexagonal numbers in the stepped pyramid described in A245092. %C A346864 T(n,k) is also the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(A014105(n)), from the border to the center, hence the sum of the n-th row of triangle is equal to A014105(n). %C A346864 T(n,k) is also the difference between the total number of partitions of all positive integers <= n-th second hexagonal number into exactly k consecutive parts, and the total number of partitions of all positive integers <= n-th second hexagonal number into exactly k + 1 consecutive parts. %C A346864 1 together with the first column gives A317186. - _Michel Marcus_, Jan 12 2025 %e A346864 Triangle begins: %e A346864 2, 1; %e A346864 6, 2, 1, 1; %e A346864 11, 4, 3, 1, 1, 1; %e A346864 19, 6, 4, 2, 2, 1, 1, 1; %e A346864 28, 10, 5, 3, 3, 2, 1, 1, 1, 1; %e A346864 40, 13, 7, 5, 3, 2, 2, 2, 1, 1, 1, 1; %e A346864 53, 18, 10, 5, 4, 3, 3, 2, 1, 2, 1, 1, 1, 1; %e A346864 69, 23, 12, 7, 5, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1; %e A346864 86, 29, 15, 9, 6, 5, 4, 2, 3, 2, 2, 1, 2, 1, 1, 1, 1, 1; %e A346864 ... %e A346864 Illustration of initial terms: %e A346864 Column h gives the n-th second hexagonal number (A014105). %e A346864 Column S gives the sum of the divisors of the second hexagonal numbers which equals the area (and the number of cells) of the associated diagram. %e A346864 -------------------------------------------------------------------------------------- %e A346864 n h S Diagram %e A346864 -------------------------------------------------------------------------------------- %e A346864 _ _ _ _ %e A346864 | | | | | | | | %e A346864 _ _|_| | | | | | | %e A346864 1 3 4 |_ _|1 | | | | | | %e A346864 2 | | | | | | %e A346864 _ _| | | | | | %e A346864 | _ _| | | | | %e A346864 _ _|_| | | | | %e A346864 | _|1 | | | | %e A346864 _ _ _ _ _| | 1 | | | | %e A346864 2 10 18 |_ _ _ _ _ _|2 | | | | %e A346864 6 _ _ _ _|_| | | %e A346864 | | | | %e A346864 _| | | | %e A346864 | _| | | %e A346864 _ _|_| | | %e A346864 _ _| _|1 | | %e A346864 |_ _ _|1 1 | | %e A346864 | 3 _ _ _ _ _ _ _| | %e A346864 |4 | _ _ _ _ _ _| %e A346864 _ _ _ _ _ _ _ _ _ _ _| | | %e A346864 3 21 32 |_ _ _ _ _ _ _ _ _ _ _| _ _| | %e A346864 11 | | %e A346864 _| _ _| %e A346864 | | %e A346864 _ _| _| %e A346864 _ _| _| %e A346864 | _|1 %e A346864 _ _ _| _ _|1 1 %e A346864 | | 2 %e A346864 | _ _ _ _|2 %e A346864 | | 4 %e A346864 | | %e A346864 | |6 %e A346864 | | %e A346864 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | %e A346864 4 36 91 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| %e A346864 19 %e A346864 . %o A346864 (PARI) row(n) = my(m=n*(2*n + 1)); vector((sqrtint(8*m+1)-1)\2, k, ceil((m+1)/k - (k+1)/2) - ceil((m+1)/(k+1) - (k+2)/2)); \\ _Michel Marcus_, Jan 12 2025 %Y A346864 Row sums give A014105, n >= 1. %Y A346864 Row lengths give A005843. %Y A346864 For the characteristic shape of sigma(A000040(n)) see A346871. %Y A346864 For the characteristic shape of sigma(A000079(n)) see A346872. %Y A346864 For the characteristic shape of sigma(A000217(n)) see A346873. %Y A346864 For the visualization of Mersenne numbers A000225 see A346874. %Y A346864 For the characteristic shape of sigma(A000384(n)) see A346875. %Y A346864 For the characteristic shape of sigma(A000396(n)) see A346876. %Y A346864 For the characteristic shape of sigma(A008588(n)) see A224613. %Y A346864 For the characteristic shape of sigma(A174973(n)) see A317305. %Y A346864 Cf. A000203, A237591, A237593, A245092, A249351, A262626, A317186. %K A346864 nonn,tabf %O A346864 1,1 %A A346864 _Omar E. Pol_, Aug 17 2021