OFFSET
0,2
COMMENTS
Number of compositions of even natural numbers into 4 parts <=n.
Number of ways of placings of an even number of indistinguishable objects into 4 distinguishable boxes with the condition that in each box there can be at most n objects.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = 1/2*((n + 1)^4 + ((1 + (-1)^n)*1/2)^4).
a(n) = +4*a(n-1) -5*a(n-2) +5*a(n-4) -4*a(n-5) +1*a(n-6).
G.f.: (1 + 4*x + 14*x^2 + 4*x^3 + x^4)/((1 + x)*(1 - x)^5).
a(n) = (n+1)^4 - floor((n+1)^4/2). - Bruno Berselli, Jan 18 2017
EXAMPLE
a(1)=8: there are 8 compositions of even natural numbers into 4 parts <=1
(0,0,0,0);
(0,0,1,1), (0,1,0,1), (0,1,1,0), (1,0,0,1), (1,0,1,0), (1,1,0,0);
(1,1,1,1).
a(2)=41: there are 41 compositions of even natural numbers into 4 parts <=2
for 0: (0,0,0,0);
for 2: (0,0,0,2), (0,0,2,0), (0,2,0,0), (2,0,0,0), (0,0,1,1), (0,1,0,1), (0,1,1,0), (1,0,0,1), (1,0,1,0), (1,1,0,0);
for 4: (0,0,2,2), (0,2,0,2), (0,2,2,0), (2,0,0,2), (2,0,2,0), (2,2,0,0), (0,1,1,2), (0,1,2,1), (0,2,1,1), (1,0,1,2), (1,0,2,1), (1,1,0,2), (1,1,2,0), (1,2,0,1), (1,2,1,0), (2,0,1,1), (2,1,0,1), (2,1,1,0), (1,1,1,1);
for 6: (0,2,2,2), (2,0,2,2), (2,2,0,2), (2,2,2,0), (1,1,2,2), (1,2,1,2), (1,2,2,1), (2,1,1,2), (2,1,2,1), (2,2,1,1);
for 8: (2,2,2,2).
MATHEMATICA
Table[1/2((n + 1)^4 + ((1 + (-1)^n)*1/2)^4), {n, 0, 25}]
Ceiling[Range[40]^4/2] (* Bruno Berselli, Jan 18 2017 *)
PROG
(Magma) [1/2*((n+1)^4+((1+(-1)^n)*1/2)^4): n in [0..40]]; // Vincenzo Librandi, Jun 16 2011
(PARI) a(n) = ceil(n^4/2); \\ Michel Marcus, Dec 14 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Adi Dani, May 29 2011
EXTENSIONS
Better name from Enrique Pérez Herrero, Dec 14 2013
STATUS
approved