A002721 - OEIS

%I M4864 N2080 #68 Oct 20 2023 12:33:20

%S 1,12,132,847,3921,14506,45402,124707,308407,699766,1477686,2936517,

%T 5540107,9993192,17333536,29048541,47220357,74703832,115341952,

%U 174223731,257989821,375191422,536708382,756232687,1050823851,1441543026,1954172962

%N Number of 3 X 3 X 3 arrays M_ijk (1 <= i,j,k <= 3) with entries satisfying 0 <= M_ijk <= n and all line sums equal to n.

%C Number of 3 X 3 X 3 arrays M_ijk (1 <= i,j,k <= 3) satisfying Sum_i M_ijk = n (all j,k), Sum_j M_ijk = n (all i,k), Sum_k M_ijk = n (all i,j) and 0 <= M_ijk <= n.

%C The constraints imply that Sum_{i,j,k} M_ijk = 9n.

%C This is a "magic cube" in Stanley's notation (see Stanley references). - _N. J. A. Sloane_, Jul 07 2014

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, Second Edition, Section 4.6.1.

%H Vincenzo Librandi, <a href="/A002721/b002721.txt">Table of n, a(n) for n = 0..1000</a>

%H A. G. Bell, <a href="http://dx.doi.org/10.1093/comjnl/13.3.278">Partitioning integers in n dimensions</a>, The Computer Journal, 13 (1970), 278-283.

%H R. P. Stanley, <a href="/A002721/a002721.pdf">Examples of Magic Labelings</a>, Unpublished Notes, 1973. [Cached copy, with permission]

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).

%F a(n) = (1/4032) * m * (m * (m * (31 * m + 1004) + 6820) + 4272) + 1, where m = n*(n+1) (from the Bell reference). -  _Sean A. Irvine_, Jul 01 2014

%F G.f.: -(x^8+3*x^7+60*x^6+7*x^5+168*x^4+7*x^3+60*x^2+3*x+1) / (x-1)^9. - _Colin Barker_, Jul 01 2014

%e Comment from _N. J. A. Sloane_, Jul 06 2014: (Start)

%e Here are four of the twelve arrays showing that a(1) = 12 (each row shows top face, middle face, bottom face):

%e  -----------

%e  100 010 001

%e  010 001 100

%e  001 100 010

%e  -----------

%e  100 001 010

%e  010 100 001

%e  001 010 100

%e  -----------

%e  001 010 100

%e  010 100 001

%e  100 001 010

%e  -----------

%e  001 100 010

%e  010 001 100

%e  100 010 001

%e  -----------

%e Each face must show one of the six 3 X 3 permutation matrices. There are 6 choices for the top face, and for each of these there are two choices for the second face and the third face is then determined, for a total of a(1)=6*2*1=12. (End)

%p A002721:=n->(1/4032)*n*(n+1)*(n*(n+1)*(n*(n+1)*(31*n*(n+1)+1004)+6820)+ 4272)+1: seq(A002721(n), n=0..30); # _Wesley Ivan Hurt_, Jul 01 2014

%t CoefficientList[Series[-(x^8 + 3*x^7 + 60*x^6 + 7*x^5 + 168*x^4 + 7*x^3 + 60*x^2 + 3*x + 1)/(x - 1)^9, {x, 0, 30}], x] (* _Wesley Ivan Hurt_, Jul 01 2014 *)

%o (PARI) Vec(-(x^8+3*x^7+60*x^6+7*x^5+168*x^4+7*x^3+60*x^2+3*x+1)/(x-1)^9 + O(x^100)) \\ _Colin Barker_, Jul 01 2014

%o (Magma) [(1/4032)*n*(n+1)*(n*(n+1)*(n*(n+1)*(31*n*(n+1)+1004)+6820)+4272)+1 : n in [0..30] ]; // _Wesley Ivan Hurt_, Jul 01 2014

%Y See A001496 for the two-dimensional 4 X 4 analog. Cf. also A002817.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Sean A. Irvine_, Jul 01 2014

%E Edited by _N. J. A. Sloane_, Jul 06 2014