A008233 a(n) = floor(n/4)*floor((n+1)/4)*floor((n+2)/4)*floor((n+3)/4).

0, 0, 0, 0, 1, 2, 4, 8, 16, 24, 36, 54, 81, 108, 144, 192, 256, 320, 400, 500, 625, 750, 900, 1080, 1296, 1512, 1764, 2058, 2401, 2744, 3136, 3584, 4096, 4608, 5184, 5832, 6561, 7290, 8100, 9000, 10000, 11000, 12100, 13310, 14641, 15972, 17424, 19008, 20736

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

a(n) is the maximal product of four nonnegative integers whose sum is n. - Andres Cicuttin, Sep 26 2018

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..3000
• Dhruv Mubayi, Counting substructures II: Hypergraphs, preprint, 2012.
• Dhruv Mubayi, Counting substructures II: Hypergraphs, Combinatorica 33 (2013), no. 5, 591--612. MR3132928.
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,3,-6,3,0,-3,6,-3,0,1,-2,1).

Crossrefs

Maximal product of k positive integers with sum n, for k = 2..10: A002620 (k=2), A006501 (k=3), this sequence (k=4), A008382 (k=5), A008881 (k=6), A009641 (k=7), A009694 (k=8), A009714 (k=9), A354600 (k=10).

Cf. A013662.

Programs

Haskell

a008233 n = product $ map (`div` 4) [n..n+3]
-- Reinhard Zumkeller, Jun 08 2011

Magma

[Floor(n/4)*Floor((n+1)/4)*Floor((n+2)/4)*Floor((n+3)/4): n in [0..50]]; // Vincenzo Librandi, Jun 09 2011

Maple

A008233:=n->floor(n/4)*floor((n+1)/4)*floor((n+2)/4)*floor((n+3)/4); seq(A008233(n), n=0..50); # Wesley Ivan Hurt, Dec 31 2013

Mathematica

Table[Floor[n/4]*Floor[(n + 1)/4]*Floor[(n + 2)/4]*Floor[(n + 3)/4], {n, 0, 50}] (* Stefan Steinerberger, Apr 03 2006 *)
Table[Times@@Floor[Range[n,n+3]/4],{n,0,50}] (* Harvey P. Dale, Mar 30 2019 *)

PARI

a(n) = prod(i=0, 3, (n+i)\4); \\ Altug Alkan, Sep 27 2018

Formula

Let b(n) = A002620(n), the quarter-squares. Then this sequence is b(0)*b(0), b(0)*b(1), b(1)*b(1), b(1)*b(2), b(2)*b(2), b(2)*b(3), ...

From R. J. Mathar, Feb 20 2011: (Start)

a(n) = 2*a(n-1) - a(n-2) + 3*a(n-4) - 6*a(n-5) + 3*a(n-6) - 3*a(n-8) + 6*a(n-9) - 3*a(n-10) + a(n-12) - 2*a(n-13) + a(n-14).

G.f.: -x^4*(1+x^6+x^2+2*x^3+x^4) / ( (1+x)^3*(x^2+1)^3*(x-1)^5 ). (End)

Sum_{n>=4} 1/a(n) = 1 + zeta(4). - Amiram Eldar, Jan 10 2023

a(4*n) = n^4. - Bernard Schott, Jan 24 2023

Extensions

More terms from Stefan Steinerberger, Apr 03 2006