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 A130046 #24 Apr 15 2023 01:49:23 %S A130046 1,1,1,1,1,1,1,3,1,1,1,1,1,2,5,2,1,5,5,1,1,2,1,1,2,1,2,8,8,2,1,8,15,8, %T A130046 1,2,8,8,2,1,2,1,1,2,1,3,11,11,3,3,18,31,18,3,1,11,31,31,11,1,2,11,18, %U A130046 11,2,1,3,3,1,1,3,3,1,3,15,24,15,3,3,24,60,60,24,3,1,15,60,93,60,15,1,3,24 %N A130046 Hexagonal pyramid of Pascal numbers in 3 dimensions. The 3-dimensional sequence is split into slices of the pyramid which in turn consist of rows of the slice, each containing multiple columns of numbers and where each element of slice j is composed of the sum of the three elements above it in slice j-1. %C A130046 Successive slices [0,0], [1,0], [1,1], [2,1], [2,2], [3,2], [3,3], [4,3], [4,4], ... in table A109672; see also A046816 (slices [n,0]), A109673 (slices [n,n]), A109649 (slices [0,k]), A109390 (slices [n,1]), A109393 (slices [1,k]), A109494 (slices [n,2]), A109495 (slices [2,k]). - _Philippe Deléham_, May 03 2007 %H A130046 Jeffrey C. Jacobs, <a href="/A130046/b130046.txt">Table of n, a(n) for n = 0..30324</a> %H A130046 Jeffrey C. Jacobs, <a href="/A130046/a130046.py.txt">Python program for this sequence</a> %F A130046 Let j be a given slice of the hexagonal pyramid. For j = 0, there are no elements. %F A130046 For j > 0, let a[x] to a[x+y-1] represent the elements of the slice, where x is the (j-1)th element of A019298 and y is the j-th element of A077043. Each slice j consists of j rows of varying column length, numbered 0 to j-1. %F A130046 The length of the first row of slice j is given by floor((j+1)/2) and the last row by floor(j/2)+1, where by convention the last row is always greater or equal in length to the first row. %F A130046 The floor(j/2)th row is j columns in length and any row before it is given by the formula floor((j+1)/2) + row#. For rows after the floor(j/2)th row, the length is given by floor(j/2) + j - row#. %F A130046 The elements a[x] to a[x+y-1] are thus layed out as a concatenated series of rows of varying column lengths as specified above. %F A130046 Thus for a given slice j, the element at row row# and column col# is represented by a[x + floor((j+1)/2) * row# + row# * (row# - 1) / 2 + col# ] when row# <= floor(j/2) and by a[x + y - (floor(j/2) + 1) * (j - row#) - (j - row#) * (j - row# + 1) / 2 + col#] otherwise, where x and y are defined above and row# and col# start counting from 0. %F A130046 The elements of a for a given slice j, row# and col#, represented by the coordinate pair (row#, col#), is given by the following recursive relation: %F A130046 For j = 1, there is 1 element whose value is 1 at (0, 0). Call this Slice[1] whose first and only element forms a0 = 1. %F A130046 For j > 1, each element (row#, col#) is given by the sum of the 3 elements above it in the pyramid. If the preceding slice does not contain one of the cells specified because the coordinates are invalid for that slice, the value is assumed to be 0. %F A130046 The cells above can be found using the following formula for a given cell Slice[j](row#, col#): %F A130046 If j is odd: %F A130046 If row# > floor(j/2): %F A130046 Sum: %F A130046 Slice[j-1](row#, col#-1) %F A130046 Slice[j-1](row#-1,col#) %F A130046 Slice[j-1](row#-1,col#-1) %F A130046 Otherwise: %F A130046 Sum: %F A130046 Slice[j-1](row#, col#) %F A130046 Slice[j-1](row#-1,col#) %F A130046 Slice[j-1](row#-1,col#-1) %F A130046 Otherwise: %F A130046 If row# > floor(j/2): %F A130046 Sum: %F A130046 Slice[j-1](row#, col#) %F A130046 Slice[j-1](row#-1,col#) %F A130046 Slice[j-1](row#,col#-1) %F A130046 Otherwise: %F A130046 Sum: %F A130046 Slice[j-1](row#, col#) %F A130046 Slice[j-1](row#,col#-1) %F A130046 Slice[j-1](row#-1,col#-1) %F A130046 Each slice is also a solution to the Prouhet-Tarry-Escott problem for a given n and k. The slices[n,k] in sequence A109672 map to the slices here by the relation k + n = j - 1, where k = n (j odd) or k = n + 1 (j even). %F A130046 When j is even, k = n - 1 would also be a solution to the Pascal hexagonal pyramid, however here the k = n + 1 solution is chosen. For j even, the slices are also given by A109673. %F A130046 Only 3 of the 6 hexagonal vertices have corresponding cells in the slice below them. Only every other vertex has a cell below it and all vertices with cells below them are always separated by 2 edges. %F A130046 By convention, when constructing Slice[j] for j odd, the uppermost vertices of Slice[j-1] are chosen to have cells below them and for j even the 2 vertices adjacent to the uppermost vertices of Slice[j-1] are chosen. %e A130046 Slice[0]: %e A130046 ... %e A130046 Slice[1]: %e A130046 1 %e A130046 Slice[2]: %e A130046 .1 %e A130046 1.1 %e A130046 Slice[3]: %e A130046 .1.1 %e A130046 1.3.1 %e A130046 .1.1 %e A130046 Slice[4]: %e A130046 ..1.1 %e A130046 .2.5.2 %e A130046 1.5.5.1 %e A130046 .1.2.1 %e A130046 Slice[5]: %e A130046 ....1..2..1 %e A130046 ..2..8..8..2 %e A130046 .1..8.15..8..1 %e A130046 ..2..8..8..2 %e A130046 ....1..2..1 %e A130046 Slice[6]: %e A130046 .....1..2..1 %e A130046 ....3.11.11..3 %e A130046 ..3.18.31.18..3 %e A130046 .1.11.31.31.11..1 %e A130046 ..2.11.18.11..2 %e A130046 ....1..3..3..1 %Y A130046 Cf. A077043, A019298. %Y A130046 Subsequence of A109672 table of slices. %Y A130046 Other tables of slices (see 2007 comment from _Philippe Deléham_): A046816, A109390, A109393, A109494, A109495, A109649, A109673. %Y A130046 Cf. A007318 (Pascal's triangle). %K A130046 nonn %O A130046 0,8 %A A130046 Jeffrey C. Jacobs (darklord(AT)timehorse.com), May 03 2007