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%I #64 Apr 08 2024 18:36:28
%S 1,27,216,1000,3375,9261,21952,46656,91125,166375,287496,474552,
%T 753571,1157625,1728000,2515456,3581577,5000211,6859000,9261000,
%U 12326391,16194277,21024576,27000000,34328125,43243551,54010152,66923416
%N Cubes of triangular numbers: (n*(n+1)/2)^3.
%C Three-dimensional cage assemblies. (See Chapter 61, "Hyperspace Prisons", of C. A. Pickover's book "Wonders of Numbers" for full explanation of "cage numbers.")
%C 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
%C 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
%D C. A. Pickover. "Wonders of Numbers: Adventures in Mathematics, Mind and Meaning." Oxford University Press. New York, NY, 2001.
%H Harry J. Smith, <a href="/A059827/b059827.txt">Table of n, a(n) for n = 1..1000</a>
%H Mauro Fiorentini, <a href="http://www.bitman.name/math/article/84/115/">Pi, occorrenze in teoria dei numeri</a>, (in Italian).
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F a(n) = Sum_{j=1..n} Sum_{i=1..n} i*j^3. - _Alexander Adamchuk_, Jun 25 2006
%F a(n) = (A000217(n))^3. - _Zak Seidov_, Jan 21 2012
%F G.f.: x*(1 + 20*x + 48*x^2 + 20*x^3 + x^4)/(1 - x)^7. - _Colin Barker_, Apr 24 2012
%F Sum_{n>=1} 1/a(n) = 80 - 8*Pi^2 (Ramanujan). - _Jaume Oliver Lafont_, Jul 17 2017
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 96*log(2) + 12*zeta(3) -80. - _Amiram Eldar_, May 14 2022
%p for n from 1 to 100 do printf(`%d,`,((n^3)*(n + 1)^3)/8) od:
%p [seq(binomial(n+2,n)^3,n=0..50)]; # _Zerinvary Lajos_, May 17 2006
%t Table[(n(n+1)/2)^3, {n,1000}] (* _Zak Seidov_, Jan 21 2012 *)
%o (PARI) { for (n=1, 1000, write("b059827.txt", n, " ", (n*(n + 1)/2)^3); ) } \\ _Harry J. Smith_, Jun 29 2009
%Y Cf. A000217, A070307, A001844.
%Y Cf. A357178 (first differences).
%K nonn,easy,changed
%O 1,2
%A _Jason Earls_, Feb 24 2001
%E More terms from _James A. Sellers_, Feb 26 2001
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