

A059173


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


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
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OFFSET

0,2


COMMENTS

2 * A000127(n).
From Raphie Frank Nov 24 2012: (Start)
Define the gross polygonal sum, GPS(n), of an ngon as the maximal number of combined points (p), intersections (i), connections (c = edges (e) + diagonals (d)) and areas (a) of a fully connected ngon, plus the area outside the ngon. 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 (n1) of Pascal's triangle.
For example, the gross polygonal sum of a 7gon (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(n1, k) + Sum_{i=0..k} C(n, i) regions.
G.f.: (x^5 + x^4  2*x^3 + 4*x^2  3*x + 1)/(x1)^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



