A059173 Maximal number of regions into which 4-space can be divided by n hyperspheres.

1, 2, 4, 8, 16, 32, 62, 114, 198, 326, 512, 772, 1124, 1588, 2186, 2942, 3882, 5034, 6428, 8096, 10072, 12392, 15094, 18218, 21806, 25902, 30552, 35804, 41708, 48316, 55682, 63862, 72914, 82898, 93876, 105912, 119072, 133424, 149038

Offset: 0

Author

N. J. A. Sloane, Feb 15 2001

Keywords

• nonn
• easy

Comments

n hyperspheres divide R^k into at most C(n-1, k) + Sum_{i=0..k} C(n, i) regions.

From Raphie Frank Nov 24 2012: (Start)

Define the gross polygonal sum, GPS(n), of an n-gon as the maximal number of combined points (p), intersections (i), connections (c = edges (e) + diagonals (d)) and areas (a) of a fully connected n-gon, plus the area outside the n-gon. The gross polygonal sum (p + i + c + a + 1) is equal to this sequence and, for all n > 0, then individual components of this sum can be calculated from the first 5 entries in row (n-1) of Pascal's triangle.

For example, the gross polygonal sum of a 7-gon (the heptagon):

Let row 6 of Pascal's triangle = {1, 6, 15, 20, 15, 6, 1} = A B C D E F G.

Points = 1 + 6 = A + B = 7 [A000027(n)].

Intersections = 20 + 15 = D + E = 35 [A000332(n+2)].

Connections = 6 + 15 = B + C = 21 [A000217(n)].

Areas inside = 15 + 20 + 15 = C + D + E = 50 [A006522(n+1)].

Areas outside = 1 = A = 1 [A000012(n)].

Then, GPS(7) = 7 + 35 + 21 + 50 + 1 = 2(A + B + C + D + E) = 114 = a(7). In general, a(n) = GPS(n). (End)

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 73, Problem 4.

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A014206 (dim 2), A046127 (dim 3), A059173 (dim 4), A059174 (dim 5).

A row of A059250.

Cf. A000012, A000027, A000217, A000332, A006522.

Programs

Mathematica

LinearRecurrence[{5,-10,10,-5,1},2^Range[0,5],50] (* Paolo Xausa, Dec 29 2023 *)

Formula

a(0) = 1; a(n) = 2 * A000127(n), for n >= 1.

G.f.: -(x^5 + x^4 - 2*x^3 + 4*x^2 - 3*x + 1)/(x-1)^5. - Colin Barker, Oct 06 2012

E.g.f.: exp(x)*(2 + x^2 + x^4/12) - 1. - Stefano Spezia, May 19 2024