A006324 a(n) = n*(n + 1)*(2*n^2 + 2*n - 1)/6.

1, 11, 46, 130, 295, 581, 1036, 1716, 2685, 4015, 5786, 8086, 11011, 14665, 19160, 24616, 31161, 38931, 48070, 58730, 71071, 85261, 101476, 119900, 140725, 164151, 190386, 219646, 252155, 288145, 327856, 371536, 419441, 471835, 528990, 591186

Offset: 1

Author

Albert Rich (Albert_Rich(AT)msn.com), Jun 14 1998

Keywords

• nonn
• easy

Comments

4-dimensional analog of centered polygonal numbers.

Partial sums of A000447. - Zak Seidov, May 19 2006

From Johannes W. Meijer, Jun 27 2009: (Start)

Equals the absolute values of the coefficients that precede the a(n-1) factors of the recurrence relations RR(n) of A162011.

This sequence enabled the analysis of A162012 and A162013. (End)

Equals the number of integer quadruples (x,y,z,w) such that min(x,y) < min(z,w), max(x,y) < max(z,w), and 0 <= x,y,z,w <= n. - Andrew Woods, Apr 21 2014

For n>3 a(n)=twice the area of an irregular quadrilateral with vertices at the points (C(n,4),C(n+1,4)), (C(n+1,4),C(n+2,4)), (C(n+2,4),C(n+3,4)), and (C(n+3,4),C(n+4,4)). - J. M. Bergot, Jun 14 2014

Links

• Delbert L. Johnson, Table of n, a(n) for n = 1..20000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000447, A000539, A000217, A002415, A053134.

Cf. A162011, A162012, a(n-2), and A162013, a(n-3). - Johannes W. Meijer, Jun 27 2009

Programs

Magma

[ n*(n + 1)*(2*n^2 + 2*n - 1)/6 : n in [1..30] ]; // Wesley Ivan Hurt, Jun 14 2014

Maple

A006324:=n->n*(n + 1)*(2*n^2 + 2*n - 1)/6; seq(A006324(n), n=1..30); # Wesley Ivan Hurt, Jun 14 2014

Mathematica

Table[Sum[k^5,{k,n}]/Sum[k,{k,n}], {n,40}] (* Alexander Adamchuk, Apr 12 2006 *)

Formula

a(n) = 8*C(n + 2, 4) + C(n + 1, 2).

a(n) = (Sum_{k=1..n} k^5) / (Sum_{k=1..n} k) = A000539(n) / A000217(n). - Alexander Adamchuk, Apr 12 2006

From Johannes W. Meijer, Jun 27 2009: (Start)

Recurrence relation 0 = Sum_{k=0..5} (-1)^k*binomial(5,k)*a(n-k).

G.f.: (1+6*z+z^2)/(1-z)^5. (End)

a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - Wesley Ivan Hurt, May 02 2021

Sum_{n>=1} 1/a(n) = 6 + 2*sqrt(3)*Pi*tan(sqrt(3)*Pi/2). - Amiram Eldar, Aug 23 2022

a(n) = A053134(n-1) - 4*A002415(n). - Yasser Arath Chavez Reyes, Feb 12 2024

Extensions

Simpler definition from Alexander Adamchuk, Apr 12 2006

More terms from Zak Seidov