OFFSET
1,2
COMMENTS
Previous definition was: Consider a figure like this <> (a squashed square, symmetric about both axes); each side is given 1 of n colors; a(n) = number of possibilities, allowing turning over.
Also number of 2 X 2 matrices with entries mod n, up to row and column permutation. Number of k X l matrices with entries mod n, up to row and column permutation is Z(S_k X S_l; n,n,...) where Z(S_k X S_l; x_1,x_2,...) is cycle index of Cartesian product of symmetric groups S_k and S_l of degree k and l, respectively. - Vladeta Jovovic, Nov 04 2000
Also, if a 2-set Y and a 3-set Z are disjoint subsets of an n-set X then a(n-5) is the number of 6-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 08 2007
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..1000
Jean-Paul Delahaye, Le miraculeux "lemme de Burnside", pp. 145-6 in 'Pour la Science' (French edition of 'Scientific American'), No. 350, December 2006, Paris.
Milan Janjic, Two Enumerative Functions.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
From Harvey P. Dale, Oct 01 2011: (Start)
G.f.: (1 + 2*x + 2*x^2 + x^3)/(1 - x)^5.
a(1)=1, a(2)=7, a(3)=27, a(4)=76, a(5)=175; for n>5, a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). (End)
E.g.f.: x*(4 + 10*x + 6*x^2 + x^3)*exp(x)/4. - Ilya Gutkovskiy, Apr 16 2016
a(n) = t(n-1)*t(n) + t(n-1) + t(n) where t=A000217. - J. M. Bergot, Apr 16 2016
a(n) = T(T(n-1)) + T(T(n)) where T(n) = A000217(n). - Charlie Marion, Feb 09 2023
Sum_{n>=1} 1/a(n) = 2*(1 + Pi^2 - sqrt(3)*Pi*coth(sqrt(3)*Pi))/9. - Amiram Eldar, Feb 13 2023
MAPLE
MATHEMATICA
Table[(n^2 (n^2+3))/4, {n, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {1, 7, 27, 76, 175}, 40] (* Harvey P. Dale, Oct 01 2011 *)
PROG
(PARI) Vec((-1-2*x-2*x^2-x^3)/(x-1)^5 + O(x^50)) \\ Michel Marcus, Aug 23 2015
(PARI) a(n) = (1/4)*n^2*(n^2+3); \\ Altug Alkan, Apr 16 2016
(Magma) [n^2*(n^2+3)/4 : n in [1..50]]; // Wesley Ivan Hurt, Dec 26 2016
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
Christian Meland (christian.meland(AT)pfi.no)
EXTENSIONS
More terms from Sam Alexander
Simplified the definition. - N. J. A. Sloane, Apr 20 2016
STATUS
approved