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%I #38 Dec 29 2023 13:53:56
%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 hyperspheres.
%C n hyperspheres divide R^k into at most C(n-1, k) + Sum_{i=0..k} C(n, i) regions.
%C From _Raphie Frank_ Nov 24 2012: (Start)
%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 Paolo Xausa, <a href="/A059173/b059173.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(0) = 1; a(n) = 2 * A000127(n), for n >= 1.
%F G.f.: -(x^5 + x^4 - 2*x^3 + 4*x^2 - 3*x + 1)/(x-1)^5. [_Colin Barker_, Oct 06 2012]
%t LinearRecurrence[{5,-10,10,-5,1},2^Range[0,5],50] (* _Paolo Xausa_, Dec 29 2023 *)
%Y Cf. A014206 (dim 2), A046127 (dim 3), A059173 (dim 4), A059174 (dim 5).
%Y A row of A059250.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Feb 15 2001
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