A059827 Cubes of triangular numbers: (n*(n+1)/2)^3.

1, 27, 216, 1000, 3375, 9261, 21952, 46656, 91125, 166375, 287496, 474552, 753571, 1157625, 1728000, 2515456, 3581577, 5000211, 6859000, 9261000, 12326391, 16194277, 21024576, 27000000, 34328125, 43243551, 54010152, 66923416

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

Cf. A000217, A070307, A001844.

Cf. A357178 (first differences).

Sequence in context: A016767 A224354 A224013 * A117688 A272342 A107054

Adjacent sequences: A059824 A059825 A059826 * A059828 A059829 A059830

Keyword

nonn,easy

Author

Jason Earls, Feb 24 2001

Extensions

More terms from James A. Sellers, Feb 26 2001

Status

approved