|
%I #13 Oct 21 2022 21:27:11
%S 0,32,216,768,2000,4320,8232,14336,23328,36000,53240,76032,105456,
%T 142688,189000,245760,314432,396576,493848,608000,740880,894432,
%U 1070696,1271808,1500000,1757600,2047032,2370816,2731568,3132000
%N a(n) = 4*(n^4-n^3).
%C a(n) is the number of edges in a four-dimensional hypercube (a tesseract) having sides of length n.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F O.g.f.: (32*x^2+56*x^3+8*x^4)/(1-x)^5.
%F E.g.f.: 4*exp(x)*x^2 (4 + 5 x + x^2).
%F From _Amiram Eldar_, Jan 14 2021: (Start)
%F Sum_{n>=2} 1/a(n) = 3/4 - Pi^2/24 - zeta(3)/4.
%F Sum_{n>=2} (-1)^n/a(n) = -3/4 + Pi^2/48 + log(2)/2 + 3*zeta(3)/16. (End)
%e a(1) = 32 because the four dimensional unit hypercube has 32 edges.
%t Table[4 (n^4 - n^3), {n, 20}]
%t LinearRecurrence[{5,-10,10,-5,1},{0,32,216,768,2000},30] (* _Harvey P. Dale_, Nov 05 2017 *)
%o (PARI) a(n)=4*(n^4-n^3) \\ _Charles R Greathouse IV_, Oct 21 2022
%Y Cf. A046092, A059986.
%K nonn,easy
%O 1,2
%A _Geoffrey Critzer_, May 18 2009
%E More terms from _Harvey P. Dale_, Nov 05 2017
%E Offset corrected by _Amiram Eldar_, Jan 14 2021
|
