A248598 a(n) = (2*n+23)*n*(n-1), a coefficient appearing in the formula a(n)*Pi/324+n+1 giving the average number of regions into which n random planes divide the cube.

0, 0, 54, 174, 372, 660, 1050, 1554, 2184, 2952, 3870, 4950, 6204, 7644, 9282, 11130, 13200, 15504, 18054, 20862, 23940, 27300, 30954, 34914, 39192, 43800, 48750, 54054, 59724, 65772, 72210, 79050, 86304, 93984, 102102, 110670, 119700

Offset: 0

Author

Jean-François Alcover, Oct 09 2014

Keywords

• nonn
• easy

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).

Programs

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 *)

PARI

concat([0,0], Vec(-6*x^2*(7*x-9)/(x-1)^4 + O(x^100))) \\ Colin Barker, Oct 09 2014

Formula

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Colin Barker, Oct 09 2014

G.f.: -6*x^2*(7*x-9) / (x-1)^4. - Colin Barker, Oct 09 2014