A208954 a(n) = n^4*(n-1)*(n+1)/12.

0, 4, 54, 320, 1250, 3780, 9604, 21504, 43740, 82500, 146410, 247104, 399854, 624260, 945000, 1392640, 2004504, 2825604, 3909630, 5320000, 7130970, 9428804, 12313004, 15897600, 20312500, 25704900, 32240754, 40106304, 49509670, 60682500, 73881680, 89391104

Offset: 1

Author

J. M. Bergot, May 31 2012

Keywords

• nonn
• easy

Comments

The product of a 2 X n matrix and a n X 2 matrix will give a constant result when the entries are the same consecutive numbers for each matrix. These constants for n are listed in the sequence.

Let k be the least consecutive number for the entries in a 2xn matrix. The first row will have entries k, k+1, k+2...k+n-1 and the second row k+n, k+n+1, k+n+2 ...k+2*n-1;

Its n X 2 matrix will have its first column the first row of the 2 X n and its second column the second row. The product will yield a 2x2 determinant having a value of (n^4)*(n-1)*(n+1)/12 (I thank Professor Daniel Cass for deducing this formula from the data presented.)

Examples

			For n=4 and k=-3 you get the 2x4 matrix with first row -3,-2,-1,0 and the second row 1,2,3,4.  Multiplying it by its 4x2 matrix will give 320.  If n=4 and k=151, the same 320 results.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Programs

Mathematica

Table[(n^4 (n-1)(n+1))/12,{n,40}] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{0,4,54,320,1250,3780,9604},40] (* Harvey P. Dale, Oct 01 2013 *)

Formula

G.f.: -2*x^2*(1+x)*(2*x^2+11*x+2) / (x-1)^7 . - R. J. Mathar, Dec 17 2012

From Amiram Eldar, Jan 10 2022: (Start)

Sum_{n>=2} 1/a(n) = 33 - 2*Pi^2 - 2*Pi^4/15.

Sum_{n>=2} (-1)^n/a(n) = 7*Pi^4/60 + Pi^2 - 21. (End)