A002721 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.

1, 12, 132, 847, 3921, 14506, 45402, 124707, 308407, 699766, 1477686, 2936517, 5540107, 9993192, 17333536, 29048541, 47220357, 74703832, 115341952, 174223731, 257989821, 375191422, 536708382, 756232687, 1050823851, 1441543026, 1954172962

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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.

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

This is a "magic cube" in Stanley's notation (see Stanley references). - N. J. A. Sloane, Jul 07 2014

Examples

			Comment from _N. J. A. Sloane_, Jul 06 2014: (Start)
Here are four of the twelve arrays showing that a(1) = 12 (each row shows top face, middle face, bottom face):
 -----------
 100 010 001
 010 001 100
 001 100 010
 -----------
 100 001 010
 010 100 001
 001 010 100
 -----------
 001 010 100
 010 100 001
 100 001 010
 -----------
 001 100 010
 010 001 100
 100 010 001
 -----------
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)
		

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, Second Edition, Section 4.6.1.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• A. G. Bell, Partitioning integers in n dimensions, The Computer Journal, 13 (1970), 278-283.
• R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973. [Cached copy, with permission]
• Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Crossrefs

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

Programs

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

Maple

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

Mathematica

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 *)

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

Formula

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

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

Extensions

More terms from Sean A. Irvine, Jul 01 2014

Edited by N. J. A. Sloane, Jul 06 2014