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A059174
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Maximal number of regions into which 5-space can be divided by n hyperspheres.
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8
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1, 2, 4, 8, 16, 32, 64, 126, 240, 438, 764, 1276, 2048, 3172, 4760, 6946, 9888, 13770, 18804, 25232, 33328, 43400, 55792, 70886, 89104, 110910, 136812, 167364, 203168, 244876, 293192, 348874, 412736, 485650, 568548, 662424, 768336, 887408, 1020832, 1169870
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OFFSET
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0,2
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COMMENTS
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n hyperspheres divide R^k into at most binomial(n-1, k) + Sum_{i=0..k} binomial(n, i) regions.
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 73, Problem 4.
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LINKS
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FORMULA
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a(n) = binomial(n-1, 5) + Sum_{i=0..5} binomial(n, i).
G.f.: (x^6 + 3*x^4 - 6*x^3 + 7*x^2 - 4*x + 1)/(x - 1)^6. - Colin Barker, Oct 06 2012
a(n) = 2*A006261(n-1), for n > 0. - Günter Rote, Dec 18 2018, by elementary manipulations.
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MAPLE
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seq(coeff(series((x^6+3*x^4-6*x^3+7*x^2-4*x+1)/(1-x)^6, x, n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Dec 18 2018
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MATHEMATICA
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Join[{1}, Table[((n^5 - 5 n^4 + 25 n^3 + 5 n^2 + 94 n + 120) / 60), {n, 0, 50}]] (* Vincenzo Librandi, Dec 21 2018 *)
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PROG
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(PARI) a(n) = binomial(n-1, 5) + sum(i=0, 5, binomial(n, i)); \\ Michel Marcus, Jan 29 2016
(GAP) Concatenation([1], List([1..40], n-> Binomial(n-1, 5) + Sum([0..5], i-> Binomial(n, i)))); # Muniru A Asiru, Dec 18 2018
(Magma) [1] cat [(n^5-5*n^4+25*n^3+5*n^2+94*n+120)/60: n in [0..40]]; // Vincenzo Librandi, Dec 21 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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