A105374 a(n) = 4*n^3 + 4*n.

0, 8, 40, 120, 272, 520, 888, 1400, 2080, 2952, 4040, 5368, 6960, 8840, 11032, 13560, 16448, 19720, 23400, 27512, 32080, 37128, 42680, 48760, 55392, 62600, 70408, 78840, 87920, 97672, 108120, 119288, 131200, 143880, 157352, 171640, 186768, 202760, 219640, 237432

Offset: 0

Author

Henry Bottomley, Apr 02 2005

Keywords

• easy
• nonn

Comments

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

Examples

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Essentially row or column of A102728 and A105374.

Cf. A002522, A006003, A008586, A034262.

Programs

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

Mathematica

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 *)

PARI

a(n)=4*n^3+4*n \\ Charles R Greathouse IV, Oct 16 2015

Formula

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

G.f.: 8*x*(1 + x + x^2)/(1-x)^4. - Colin Barker, May 24 2012

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 26 2012

a(n) = 8*A006003(n). - Bruce J. Nicholson, Apr 18 2017

From Elmo R. Oliveira, Aug 07 2025: (Start)

E.g.f.: 4*x*(1 + x)*(2 + x)*exp(x).

a(n) = 4*A034262(n). (End)