A008233 - OEIS

%I #68 Jan 25 2023 13:22:31

%S 0,0,0,0,1,2,4,8,16,24,36,54,81,108,144,192,256,320,400,500,625,750,

%T 900,1080,1296,1512,1764,2058,2401,2744,3136,3584,4096,4608,5184,5832,

%U 6561,7290,8100,9000,10000,11000,12100,13310,14641,15972,17424,19008,20736

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

%C a(n) is the maximal product of four nonnegative integers whose sum is n. - _Andres Cicuttin_, Sep 26 2018

%H Vincenzo Librandi, <a href="/A008233/b008233.txt">Table of n, a(n) for n = 0..3000</a>

%H Dhruv Mubayi, <a href="http://homepages.math.uic.edu/~mubayi/papers/hypcountrevHP.pdf">Counting substructures II: Hypergraphs</a>, preprint, 2012.

%H Dhruv Mubayi, <a href="http://dx.doi.org/10.100/s00493-013-2638-2">Counting substructures II: Hypergraphs</a>, Combinatorica 33 (2013), no. 5, 591--612. MR3132928.

%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,3,-6,3,0,-3,6,-3,0,1,-2,1).

%F 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), ...

%F From _R. J. Mathar_, Feb 20 2011: (Start)

%F 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).

%F 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)

%F Sum_{n>=4} 1/a(n) = 1 + zeta(4). - _Amiram Eldar_, Jan 10 2023

%F a(4*n) = n^4. - _Bernard Schott_, Jan 24 2023

%p 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

%t Table[Floor[n/4]*Floor[(n + 1)/4]*Floor[(n + 2)/4]*Floor[(n + 3)/4], {n, 0, 50}] (* _Stefan Steinerberger_, Apr 03 2006 *)

%t Table[Times@@Floor[Range[n,n+3]/4],{n,0,50}] (* _Harvey P. Dale_, Mar 30 2019 *)

%o (Haskell)

%o a008233 n = product $ map (`div` 4) [n..n+3]

%o -- _Reinhard Zumkeller_, Jun 08 2011

%o (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

%o (PARI) a(n) = prod(i=0, 3, (n+i)\4); \\ _Altug Alkan_, Sep 27 2018

%Y 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).

%Y Cf. A013662.

%K nonn,nice,easy

%O 0,6

%A _N. J. A. Sloane_

%E More terms from _Stefan Steinerberger_, Apr 03 2006