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A059173 Maximal number of regions into which 4-space can be divided by n hyper-spheres. 7

%I

%S 1,2,4,8,16,32,62,114,198,326,512,772,1124,1588,2186,2942,3882,5034,

%T 6428,8096,10072,12392,15094,18218,21806,25902,30552,35804,41708,

%U 48316,55682,63862,72914,82898,93876,105912,119072,133424,149038

%N Maximal number of regions into which 4-space can be divided by n hyper-spheres.

%C 2 * A000127(n).

%C From _Raphie Frank_ Nov 24 2012, (Begin)

%C 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.

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

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

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

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

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

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

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

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

%C (end)

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

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

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

%F G.f.: -(x^5+x^4-2*x^3+4*x^2-3*x+1)/(x-1)^5. [_Colin Barker_, Oct 06 2012]

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

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Feb 15 2001

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Last modified November 22 16:31 EST 2019. Contains 329396 sequences. (Running on oeis4.)