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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.
2
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
Eric Weisstein's MathWorld, Square Division by Lines
Eric Weisstein's MathWorld, Cube Division by Planes
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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved