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 A059250 Square array read by antidiagonals: T(k,n) = binomial(n-1, k) + Sum_{i=0..k} binomial(n, i), k >= 1, n >= 0. 7
 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, 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, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS T(k,n) = maximal number of regions into which k-space can be divided by n hyper-spheres (k >= 1, n >= 0). For all fixed k, the sequences T(k,n) are complete. - Frank M Jackson, Jan 26 2012 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 REFERENCES L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 73, Problem 4. LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened K. E. Morrison, From bocce to positivity: some probabilistic linear algebra, Math. Mag., 86 (2013) 110-119. L. SchlĂ¤fli, Theorie der vielfachen KontinuitĂ¤t, 1901. (See p. 41) J. G. Wendel, A problem in geometric probability, Math. Scand., 11 (1962) 109-111. FORMULA T(k,n) =  2 * Sum_{i=0..k-1} binomial(n-1, i), k >= 1, n >= 1. - Kent E. Morrison, Nov 11 2017 EXAMPLE Array begins   1, 2, 4, 6,  8, 10, 12, ...   1, 2, 4, 8, 14, 22, ...   1, 2, 4, 8, 16, ... MATHEMATICA 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 *) CROSSREFS Cf. A014206 (dim 2), A046127 (dim 3), A059173 (dim 4), A059174 (dim 5). Apart from border, same as A059214. If the k=0 row is included, same as A178522. Sequence in context: A327844 A243851 A168266 * A303696 A131074 A059268 Adjacent sequences:  A059247 A059248 A059249 * A059251 A059252 A059253 KEYWORD nonn,tabl AUTHOR N. J. A. Sloane, Feb 15 2001 EXTENSIONS Corrected and edited by N. J. A. Sloane, Aug 31 2011, following a suggestion from Frank M Jackson STATUS approved

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Last modified August 14 18:13 EDT 2022. Contains 356122 sequences. (Running on oeis4.)