OFFSET
0,3
COMMENTS
The analog formula giving the average number of regions into which n random lines divide the square is n*(n-1)*Pi/16+n+1.
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.1 Geometric probability constants, p. 482.
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..1000
Eric Weisstein's MathWorld, Square Division by Lines
Eric Weisstein's MathWorld, Cube Division by Planes
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
From Colin Barker, Oct 09 2014: (Start)
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4).
G.f.: -6*x^2*(7*x-9) / (x-1)^4. (End)
E.g.f.: exp(x)*x^2*(27 + 2*x). - Stefano Spezia, Sep 28 2025
MATHEMATICA
a[n_] := (2*n+23)*n*(n-1); Table[a[n], {n, 0, 40}]
LinearRecurrence[{4, -6, 4, -1}, {0, 0, 54, 174}, 50] (* Harvey P. Dale, Mar 17 2022 *)
PROG
(PARI) concat([0, 0], Vec(-6*x^2*(7*x-9)/(x-1)^4 + O(x^100))) \\ Colin Barker, Oct 09 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jean-François Alcover, Oct 09 2014
STATUS
approved