A046127 Maximal number of regions into which space can be divided by n spheres.

0, 2, 4, 8, 16, 30, 52, 84, 128, 186, 260, 352, 464, 598, 756, 940, 1152, 1394, 1668, 1976, 2320, 2702, 3124, 3588, 4096, 4650, 5252, 5904, 6608, 7366, 8180, 9052, 9984, 10978, 12036, 13160, 14352, 15614, 16948, 18356, 19840, 21402, 23044

Offset: 0

Author

Eric W. Weisstein

Keywords

• nonn
• easy
• nice

Comments

If Y is a 2-subset of an n-set X then, for n >= 2, a(n-2) is equal to the number of 2-subsets and 4-subsets of X having exactly one element in common with Y. - Milan Janjic, Dec 28 2007

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 73, Problem 4.
• A. M. Yaglom and I. M. Yaglom: Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory. New York: Dover Publications, Inc., 1987, p. 13, #45 (First published: San Francisco: Holden-Day, Inc., 1964).

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Mark de Rooij, Dion Woestenburg, and Frank Busing, Supervised and Unsupervised Mapping of Binary Variables: A proximity perspective, arXiv:2402.07624 [stat.CO], 2024. See p. 33.
• Eric Weisstein's World of Mathematics, Space Division by Spheres.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A014206 (dim 2), this sequence (dim 3), A059173 (dim 4), A059174 (dim 5). See also A000124, A000125. A row of A059250.

Cf. A033547.

Programs

Mathematica

Join[{0},Table[n (n^2-3n+8)/3,{n,50}]]  (* Harvey P. Dale, Apr 21 2011 *)

Python

def a(n): return n*(n**2 - 3*n + 8)//3 # Philip C. Ritchey, Dec 10 2017

Formula

a(n) = f(n,3) where f(n,k) = C(n-1, k) + Sum_{i=0..k} C(n, i) for hyperspheres in R^k.

a(n) = n*(n^2 - 3*n + 8)/3.

From Philip C. Ritchey, Dec 09 2017: (Start)

The above identity proved as closed form of the following summation and its corresponding recurrence relation:

a(n) = Sum_{i=1..n} (i*(i-3) + 4).

a(n) = a(n-1) + n*(n-3) + 4, a(0) = 0. (End)

From Colin Barker, Jan 28 2012: (Start)

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

G.f.: 2*x*(1 - 2*x + 2*x^2)/(1 - x)^4. (End)

a(n) = A033547(n-1) + 2 for n >= 1. - Jianing Song, Feb 03 2024

E.g.f.: exp(x)*x*(6 + x^2)/3. - Stefano Spezia, Feb 15 2024