OFFSET
1,2
COMMENTS
4-dimensional analog of centered polygonal numbers.
Partial sums ofA000447.-Zak Seidov,May 19 2006
FromJohannes 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) ofA162011.
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).
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
FromJohannes 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
MAPLE
MATHEMATICA
Table[Sum[k^5, {k, n}]/Sum[k, {k, n}], {n, 40}] (*Alexander Adamchuk,Apr 12 2006 *)
PROG
(Magma) [ n*(n + 1)*(2*n^2 + 2*n - 1)/6: n in [1..30] ]; //Wesley Ivan Hurt,Jun 14 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Albert Rich (Albert_Rich(AT)msn.com), Jun 14 1998
EXTENSIONS
Simpler definition fromAlexander Adamchuk,Apr 12 2006
More terms fromZak Seidov
STATUS
approved