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A059173 Maximal number of regions into which 4-space can be divided by n hyper-spheres. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

2 * A000127(n).

From Raphie Frank Nov 24 2012, (Begin)

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

Table of n, a(n) for n=0..38.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

FORMULA

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

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

CROSSREFS

Cf. A014206 (dim 2), A046127 (dim 3), A059173 (dim 4), A059174 (dim 5). A row of A059250.

Sequence in context: A005309 A078389 A248847 * A274005 A027560 A135493

Adjacent sequences:  A059170 A059171 A059172 * A059174 A059175 A059176

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Feb 15 2001

STATUS

approved

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Last modified August 24 04:58 EDT 2017. Contains 291052 sequences.