OFFSET
1,2
COMMENTS
Three-dimensional cage assemblies. (See Chapter 61, "Hyperspace Prisons", of C. A. Pickover's book "Wonders of Numbers" for full explanation of "cage numbers.")
For n>=0 the number of 3 X 3 matrices with nonnegative integer entries such that every row sum equals n is a(n+1). - Sharon Sela (sharonsela(AT)hotmail.com), May 14 2002
a(n) also gives the value for the number of possible cuboids (including cubes) that will fit inside an n*n*n cube. - Alexander Craggs, Mar 08 2017
REFERENCES
C. A. Pickover. "Wonders of Numbers: Adventures in Mathematics, Mind and Meaning." Oxford University Press. New York, NY, 2001.
LINKS
Harry J. Smith, Table of n, a(n) for n = 1..1000
Mauro Fiorentini, Pi, occorrenze in teoria dei numeri, (in Italian).
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
FORMULA
a(n) = Sum_{j=1..n} Sum_{i=1..n} i*j^3. - Alexander Adamchuk, Jun 25 2006
a(n) = (A000217(n))^3. - Zak Seidov, Jan 21 2012
G.f.: x*(1 + 20*x + 48*x^2 + 20*x^3 + x^4)/(1 - x)^7. - Colin Barker, Apr 24 2012
Sum_{n>=1} 1/a(n) = 80 - 8*Pi^2 (Ramanujan). - Jaume Oliver Lafont, Jul 17 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = 96*log(2) + 12*zeta(3) -80. - Amiram Eldar, May 14 2022
MAPLE
for n from 1 to 100 do printf(`%d, `, ((n^3)*(n + 1)^3)/8) od:
[seq(binomial(n+2, n)^3, n=0..50)]; # Zerinvary Lajos, May 17 2006
MATHEMATICA
Table[(n(n+1)/2)^3, {n, 1000}] (* Zak Seidov, Jan 21 2012 *)
PROG
(PARI) { for (n=1, 1000, write("b059827.txt", n, " ", (n*(n + 1)/2)^3); ) } \\ Harry J. Smith, Jun 29 2009
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jason Earls, Feb 24 2001
EXTENSIONS
More terms from James A. Sellers, Feb 26 2001
STATUS
approved