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

0,3

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

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

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

Sequence in context: A044767 A250792 A044980 * A157428 A187299 A288626

Adjacent sequences: A248595 A248596 A248597 * A248599 A248600 A248601

Keyword

nonn,easy

Author

Jean-François Alcover, Oct 09 2014

Status

approved