A308305 a(n) = s(n,n) + s(n,n-1) + s(n,n-2), where s(n,k) are the unsigned Stirling numbers of the first kind (see A132393).
1, 2, 6, 18, 46, 101, 197, 351, 583, 916, 1376, 1992, 2796, 3823, 5111, 6701, 8637, 10966, 13738, 17006, 20826, 25257, 30361, 36203, 42851, 50376, 58852, 68356, 78968, 90771, 103851, 118297, 134201, 151658, 170766, 191626, 214342, 239021, 265773, 294711
Offset: 1
References
- T. Zaslavsky, Perpendicular dissections of space. Discrete Comput. Geom. 27 (2002), no. 3, 303-351.
Links
- Colin Barker, Table of n, a(n) for n = 1..1000
- Alvaro Carbonero, Beth Anne Castellano, Gary Gordon, Charles Kulick, Karie Schmitz, and Brittany Shelton, Permutations of point sets in R_d, arXiv:2106.14140 [math.CO], 2021.
- I. J. Good and T. N. Tideman, Stirling numbers and a geometric structure from voting theory, J. Combinatorial Theory Ser. A 23 (1977), 34-45.
- T. Zaslavsky, Perpendicular dissections of space, arXiv:1001.4435 [math.CO], 2010. See equation (1.1) with d=2.
- Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
Crossrefs
The unsigned Stirling numbers of the first kind s(n,k) are given in A132393.
The division of space formulation can be generalized to higher dimensions with use of A008275 by Good and Tideman's work.
The maximum number of regions generated by pairwise perpendicular bisectors on a sphere is given by A087645.
Programs
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Magma
[(1/24)*(24 - 14*n + 21*n^2 - 10*n^3 + 3*n^4): n in [1..40]]; // Vincenzo Librandi, Jun 30 2019
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Mathematica
Table[(1/24)(24 - 14 i + 21 i^2 - 10 i^3 + 3 i^4), {i, 40}] -
PARI
Vec(x*(1 - 3*x + 6*x^2 - 2*x^3 + x^4) / (1 - x)^5 + O(x^40)) \\ Colin Barker, Jun 30 2019
Formula
a(n) = s(n,n) + s(n,n-1) + s(n,n-2), where s(n,k) are the unsigned Stirling numbers of the first kind.
a(n) = (1/24)*(24 - 14*n + 21*n^2 - 10*n^3 + 3*n^4).
From Colin Barker, Jun 30 2019: (Start)
G.f.: x*(1 - 3*x + 6*x^2 - 2*x^3 + x^4) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.
(End)
Comments