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A006324 a(n) = n*(n + 1)*(2*n^2 + 2*n - 1)/6. 10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

4-dimensional analog of centered polygonal numbers.

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

Equals the absolute values of the coefficients that precede the a(n-1) factors of the recurrence relations RR(n) of A162011. - Johannes W. Meijer, Jun 27 2009

This sequence enabled the analysis of A162012 and A162013. - Johannes W. Meijer, Jun 27 2009

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

Table of n, a(n) for n=1..36.

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

Recurrence relation 0 = sum(k=0..5, (-1)^k*binomial(5,k)*a(n-k) ). - Johannes W. Meijer, Jun 27 2009

G.f.: (1+6*z+z^2)/(1-z)^5. - Johannes W. Meijer, Jun 27 2009

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 *)

PROG

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

CROSSREFS

Cf. A000447, A000539, A000217.

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

Sequence in context: A143059 A224142 A155014 * A256582 A231887 A177370

Adjacent sequences:  A006321 A006322 A006323 * A006325 A006326 A006327

KEYWORD

nonn,easy

AUTHOR

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

EXTENSIONS

Simpler definition from Alexander Adamchuk, Apr 12 2006

More terms from Zak Seidov

STATUS

approved

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Last modified August 23 06:19 EDT 2017. Contains 290958 sequences.