login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A059250 Square array read by antidiagonals: T(k,n) = binomial(n-1, k) + Sum_{i=0..k} binomial(n, i), k >= 1, n >= 0. 6
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. Schläfli, Theorie der vielfachen Kontinuität, 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: A141539 A243851 A168266 * A131074 A059268 A256009

Adjacent sequences:  A059247 A059248 A059249 * A059251 A059252 A059253

KEYWORD

nonn,tabl,changed

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 17 22:33 EST 2017. Contains 296124 sequences.