OFFSET
0,2
COMMENTS
n hyperspheres divide R^k into at most C(n-1, k) + Sum_{i=0..k} C(n, i) regions.
From Raphie Frank Nov 24 2012: (Start)
Define the gross polygonal sum, GPS(n), of an n-gon as the maximal number of combined points (p), intersections (i), connections (c = edges (e) + diagonals (d)) and areas (a) of a fully connected n-gon, plus the area outside the n-gon. The gross polygonal sum (p + i + c + a + 1) is equal to this sequence and, for all n > 0, then individual components of this sum can be calculated from the first 5 entries in row (n-1) of Pascal's triangle.
For example, the gross polygonal sum of a 7-gon (the heptagon):
Let row 6 of Pascal's triangle = {1, 6, 15, 20, 15, 6, 1} = A B C D E F G.
Points = 1 + 6 = A + B = 7 [A000027(n)].
Intersections = 20 + 15 = D + E = 35 [A000332(n+2)].
Connections = 6 + 15 = B + C = 21 [A000217(n)].
Areas inside = 15 + 20 + 15 = C + D + E = 50 [A006522(n+1)].
Areas outside = 1 = A = 1 [A000012(n)].
Then, GPS(7) = 7 + 35 + 21 + 50 + 1 = 2(A + B + C + D + E) = 114 = a(7). In general, a(n) = GPS(n). (End)
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 73, Problem 4.
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
a(0) = 1; a(n) = 2 * A000127(n), for n >= 1.
G.f.: -(x^5 + x^4 - 2*x^3 + 4*x^2 - 3*x + 1)/(x-1)^5. - Colin Barker, Oct 06 2012
E.g.f.: exp(x)*(2 + x^2 + x^4/12) - 1. - Stefano Spezia, May 19 2024
MATHEMATICA
LinearRecurrence[{5, -10, 10, -5, 1}, 2^Range[0, 5], 50] (* Paolo Xausa, Dec 29 2023 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Feb 15 2001
STATUS
approved