OFFSET
0,2
COMMENTS
If Y is a 2-subset of an n-set X then, for n >= 2, a(n-2) is equal to the number of 2-subsets and 4-subsets of X having exactly one element in common with Y. - Milan Janjic, Dec 28 2007
Yaglom and Yaglom, pp. 102-106, implicitly suggest the following construction: draw two partially overlapping spheres of radius 1 with centers at A and B say, then draw n-2 further spheres of radius 1 at n-2 equally-spaced points along the line joining A and B.
According to Wendel's theorem, if n points are selected independently at random from a centrally symmetric 4-dimensional distribution, then a(n)/2^n is the probability that their convex hull does not contain the origin. - Amiram Eldar, Jul 13 2026
REFERENCES
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 73, Problem 4.
Răzvan Gelca and Titu Andreescu, Putnam and Beyond, Springer 2007, pp. 288, 734-735.
A. M. Yaglom and I. M. Yaglom, Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory. New York: Dover Publications, Inc., 1987, p. 13, #45; solutions pp. 102-107 (First published: San Francisco: Holden-Day, Inc., 1964).
LINKS
T. D. Noe, Table of n, a(n) for n = 0..1000
David O. H. Cutler, Jonas Karlsson, and Neil J. A. Sloane, Cutting a Pancake with an Exotic Knife, arXiv:2511.15864[math.CO], v3, April 19 2026.
Mark de Rooij, Dion Woestenburg, and Frank Busing, Supervised and Unsupervised Mapping of Binary Variables: A proximity perspective, arXiv:2402.07624 [stat.CO], 2024. See p. 33.
Eric Weisstein's World of Mathematics, Space Division by Spheres.
James G. Wendel, A Problem in Geometric Probability, Math. Scand., Vol. 11 (1962), pp. 109-111.
Wikipedia, Wendel's theorem.
A. M. Yaglom and I. M. Yaglom, Challenging Mathematical Problems with Elementary Solutions. Vol. I, Annotated scan of pp. 102-103.
A. M. Yaglom and I. M. Yaglom, Challenging Mathematical Problems with Elementary Solutions. Vol. I, Annotated scan of pp. 104-105.
A. M. Yaglom and I. M. Yaglom, Challenging Mathematical Problems with Elementary Solutions. Vol. I, Annotated scan of pp. 106-107.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
a(n) = f(n,3) where f(n,k) = C(n-1, k) + Sum_{i=0..k} C(n, i) for hyperspheres in R^k.
a(n) = n*(n^2 - 3*n + 8)/3.
From Philip C. Ritchey, Dec 09 2017: (Start)
The above identity proved as closed form of the following summation and its corresponding recurrence relation:
a(n) = Sum_{i=1..n} (i*(i-3) + 4).
a(n) = a(n-1) + n*(n-3) + 4, a(0) = 0. (End)
From Colin Barker, Jan 28 2012: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: 2*x*(1 - 2*x + 2*x^2)/(1 - x)^4. (End)
a(n) = A033547(n-1) + 2 for n >= 1. - Jianing Song, Feb 03 2024
E.g.f.: exp(x)*x*(6 + x^2)/3. - Stefano Spezia, Feb 15 2024
MATHEMATICA
Join[{0}, Table[n (n^2-3n+8)/3, {n, 50}]] (* Harvey P. Dale, Apr 21 2011 *)
PROG
(Python)
def a(n): return n*(n**2 - 3*n + 8)//3 # Philip C. Ritchey, Dec 10 2017
CROSSREFS
KEYWORD
nonn,easy,nice,changed
AUTHOR
EXTENSIONS
Definition of a(0) changed by N. J. A. Sloane, Nov 12 2025
STATUS
approved