login
a(n) = 4*n^3 + 4*n.
6

%I #33 Sep 08 2022 08:45:17

%S 0,8,40,120,272,520,888,1400,2080,2952,4040,5368,6960,8840,11032,

%T 13560,16448,19720,23400,27512,32080,37128,42680,48760,55392,62600,

%U 70408,78840,87920,97672,108120,119288,131200,143880,157352,171640,186768

%N a(n) = 4*n^3 + 4*n.

%C For n > 1, the number of straight lines with n points in a 4-dimensional hypercube of with n points on each edge is 4n^3 + 12n^2 + 16n + 8, i.e., A105374(n+1).

%H Vincenzo Librandi, <a href="/A105374/b105374.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = A002522(n)*A008586(n).

%F G.f.: 8*x*(1 + x + x^2)/(1-x)^4. - _Colin Barker_, May 24 2012

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, Jun 26 2012

%F a(n) = 8* A006003(n). - _Bruce J. Nicholson_, Apr 18 2017

%e a(5) = 4*5^3 + 4*5 = 500 + 20 = 520.

%t CoefficientList[Series[8*x*(1+x+x^2)/(1-x)^4,{x,0,40}],x] (* or *) LinearRecurrence[{4,-6,4,-1},{0,8,40,120},50] (* _Vincenzo Librandi_, Jun 26 2012 *)

%o (Magma) I:=[0, 8, 40, 120]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..45]]; // _Vincenzo Librandi_, Jun 26 2012

%o (PARI) a(n)=4*n^3+4*n \\ _Charles R Greathouse IV_, Oct 16 2015

%Y Essentially row or column of A102728 and A105374.

%Y Cf. A006003.

%K easy,nonn

%O 0,2

%A _Henry Bottomley_, Apr 02 2005