A059174 Maximal number of regions into which 5-space can be divided by n hyperspheres.

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

Offset: 0

Author

N. J. A. Sloane, Feb 15 2001

Keywords

• nonn
• easy

Comments

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

References

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

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A014206 (dim 2), A046127 (dim 3), A059173 (dim 4), this sequence (dim 5).

Fifth row (k=5) of A059250.

Cf. A006261.

Programs

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

Maple

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

Mathematica

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 *)

PARI

a(n) = binomial(n-1, 5) + sum(i=0, 5, binomial(n, i)); \\ Michel Marcus, Jan 29 2016

Formula

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.

E.g.f.: 1 + (1/60)*(120*x + 20*x^3 + x^5)*exp(x). - Franck Maminirina Ramaharo, Dec 21 2018