OFFSET
1,2
COMMENTS
Number of 5-dimensional cage assemblies.
See Chap. 61, "Hyperspace Prisons", of C. Pickover's book "Wonders of Numbers" for full explanation of "cage numbers."
REFERENCES
Clifford A. Pickover, Wonders of Numbers, Oxford University Press, 2001, p. 325.
LINKS
Harry J. Smith, Table of n, a(n) for n = 1..1000
Clifford A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
FORMULA
L(n) = ((n^m)(n + 1)^m)/(2^m) where m is the dimension.
G.f.: x * (x^8 +232*x^7 +5158*x^6+ 27664*x^5 +47290*x^4 +27664*x^3 +5158*x^2 +232*x +1) / (1-x)^11. - Colin Barker, Jun 28 2012
From Amiram Eldar, May 15 2022: (Start)
Sum_{n>=1} 1/a(n) = 4032 - 1120*Pi^2/3 - 32*Pi^4/9.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4480*log(2) + 720*zeta(3) + 60*zeta(5) - 4032. (End)
MAPLE
for n from 1 to 100 do printf(`%d, `, ((n^5)*(n + 1)^5)/(2^5) ) od:
with (combinat):seq(mul(stirling2(n+1, n), k=1..5), n=1..21); # Zerinvary Lajos, Dec 14 2007
MATHEMATICA
m = 5; Table[ ( (n^m)(n + 1)^m )/(2^m), {n, 1, 26} ]
Table[Binomial[n+1, 2]^5, {n, 20}] (* Harvey P. Dale, May 04 2018 *)
PROG
(PARI) { for (n=1, 1000, write("b059860.txt", n, " ", (n*(n + 1)/2)^5); ) } \\ Harry J. Smith, Jun 29 2009
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jason Earls, Feb 28 2001
EXTENSIONS
More terms from James A. Sellers, Feb 28 2001
Better definition from Zerinvary Lajos, May 23 2006
Corrected the definition from binomial(n+2,2)^5 to binomial(n+1,2)^5. - Harry J. Smith, Jun 29 2009
STATUS
approved