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 A059250 #54 Dec 28 2023 14:49:55
%S A059250 1,1,2,1,2,4,1,2,4,6,1,2,4,8,8,1,2,4,8,14,10,1,2,4,8,16,22,12,1,2,4,8,
%T A059250 16,30,32,14,1,2,4,8,16,32,52,44,16,1,2,4,8,16,32,62,84,58,18,1,2,4,8,
%U A059250 16,32,64,114,128,74,20,1,2,4,8,16,32,64,126,198,186,92,22,1,2,4,8,16,32,64
%N A059250 Square array read by antidiagonals: T(k,n) = binomial(n-1, k) + Sum_{i=0..k} binomial(n, i), k >= 1, n >= 0.
%C A059250 T(k,n) = maximal number of regions into which k-space can be divided by n hyperspheres (k >= 1, n >= 0).
%C A059250 For all fixed k, the sequences T(k,n) are complete. - _Frank M Jackson_, Jan 26 2012
%C A059250 T(k-1,n) is also the number of regions created by n generic hyperplanes through the origin in k-space (k >= 2). - _Kent E. Morrison_, Nov 11 2017
%D A059250 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 73, Problem 4.
%H A059250 G. C. Greubel, <a href="/A059250/b059250.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A059250 K. E. Morrison, <a href="https://arxiv.org/abs/1405.2994">From bocce to positivity: some probabilistic linear algebra</a>, arXiv:1405.2994 [math.PR], 2014; Math. Mag., 86 (2013) 110-119.
%H A059250 L. Schläfli, <a href="https://books.google.com/books?id=foIUAQAAMAAJ"> Theorie der vielfachen Kontinuität</a>, 1901. (See p. 41)
%H A059250 J. G. Wendel, <a href="http://dx.doi.org/10.7146/math.scand.a-10655">A problem in geometric probability</a>, Math. Scand., 11 (1962) 109-111.
%F A059250 T(k,n) = 2 * Sum_{i=0..k-1} binomial(n-1, i), k >= 1, n >= 1. - _Kent E. Morrison_, Nov 11 2017
%e A059250 Array begins
%e A059250 1, 2, 4, 6, 8, 10, 12, ...
%e A059250 1, 2, 4, 8, 14, 22, ...
%e A059250 1, 2, 4, 8, 16, ...
%t A059250 getvalue[n_, k_] := If[n==0, 1, Binomial[n-1, k]+Sum[Binomial[n, i],{i, 0,k}]]; lexicographicLattice[{dim_, maxHeight_}] := Flatten[Array[Sort@Flatten[(Permutations[#1] &) /@ IntegerPartitions[#1 + dim - 1, {dim}], 1] &, maxHeight], 1]; pairs=lexicographicLattice[{2, 13}]-1; Table[getvalue[First[pairs[[j]]], Last[pairs[[j]]]+1], {j, 1, Length[pairs]}] (* _Frank M Jackson_, Mar 16 2013 *)
%Y A059250 Cf. A014206 (dim 2), A046127 (dim 3), A059173 (dim 4), A059174 (dim 5).
%Y A059250 Apart from border, same as A059214. If the k=0 row is included, same as A178522.
%K A059250 nonn,tabl
%O A059250 1,3
%A A059250 _N. J. A. Sloane_, Feb 15 2001
%E A059250 Corrected and edited by _N. J. A. Sloane_, Aug 31 2011, following a suggestion from _Frank M Jackson_