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.
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 73, Problem 4.
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.
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
AUTHOR
EXTENSIONS
Definition of a(0) changed by N. J. A. Sloane, Nov 12 2025
STATUS
approved
