A364922 a(n) is the square of the side length of a simplex whose n-dimensional inner hypervolume is equal to its (n-1)-dimensional surface hypervolume. As a result, the sequence starts at n=2.

48, 216, 640, 1500, 3024, 5488, 9216, 14580, 22000, 31944, 44928, 61516, 82320, 108000, 139264, 176868, 221616, 274360, 336000, 407484, 489808, 584016, 691200, 812500, 949104, 1102248, 1273216, 1463340, 1674000, 1906624, 2162688, 2443716, 2751280, 3087000

Offset

2,1

Comments

Setting the generalized hypervolume formula equal to the surface hypervolume formula and solving for the side length x (and ignoring the x = 0 solution, as it would correspond to a simplex consisting of only a single point) gives x = sqrt(2*(n^3)*(n+1)).

Links

Harvey P. Dale, Table of n, a(n) for n = 2..1000

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = 2*n^3*(n+1) = 2*A179824(n+1).

From Stefano Spezia, Apr 13 2024: (Start)

G.f.: 4*x^2*(12 - 6*x + 10*x^2 - 5*x^3 + x^4)/(1 - x)^5.

a(n) = 4*A019582(n+1). (End)

Mathematica

Table[2*n^3*(n + 1), {n, 2, 50}] (* Paolo Xausa, Apr 18 2024 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {48, 216, 640, 1500, 3024}, 40] (* Harvey P. Dale, Aug 27 2024 *)

Prog

(Python)
def a(n): return 2 * n**3 * (n + 1)
print([a(n) for n in range(2, 50)])

Crossrefs

Cf. A019582, A179824.

Sequence in context: A183683 A260240 A260062 * A235759 A062248 A100146

Adjacent sequences: A364919 A364920 A364921 * A364923 A364924 A364925

Keyword

easy,nonn

Author

Matt Moir, Apr 13 2024

Status

approved