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, 82312875, 100544625, 122023936, 147197952

Offset: 1

Author

Jason Earls, Feb 24 2001

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000217, A070307, A001844.

Cf. A357178 (first differences).

Programs

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 *)

PARI

a(n) = { (n*(n + 1)/2)^3 } \\ Harry J. Smith, Jun 29 2009

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

E.g.f.: exp(x)*x*(8 + 100*x + 184*x^2 + 98*x^3 + 18*x^4 + x^5)/8. - Stefano Spezia, Jun 13 2025

Extensions

More terms from James Sellers, Feb 26 2001