A171714 a(n) = ceiling((n+1)^4/2).

1, 8, 41, 128, 313, 648, 1201, 2048, 3281, 5000, 7321, 10368, 14281, 19208, 25313, 32768, 41761, 52488, 65161, 80000, 97241, 117128, 139921, 165888, 195313, 228488, 265721, 307328, 353641, 405000, 461761, 524288, 592961, 668168, 750313, 839808

Offset: 0

Author

Adi Dani, May 29 2011

Keywords

• nonn
• easy

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.

Examples

			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).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A175110, A191903.

Programs

Magma

[1/2*((n+1)^4+((1+(-1)^n)*1/2)^4): n in [0..40]]; // Vincenzo Librandi, Jun 16 2011

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 *)

PARI

a(n) = ceil(n^4/2); \\ Michel Marcus, Dec 14 2013

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

Extensions

Better name from Enrique Pérez Herrero, Dec 14 2013