A173724 Number of reduced, normalized 3 X 3 semimagic squares with distinct nonnegative integer entries and maximum entry n.

1, 2, 6, 14, 21, 36, 51, 74, 93, 134, 163, 216, 248, 330, 371, 470, 526, 646, 714, 872, 942, 1130, 1224, 1440, 1543, 1810, 1922, 2220, 2363, 2700, 2853, 3252, 3409, 3854, 4053, 4536, 4744, 5304, 5525, 6134, 6396, 7056, 7330, 8080, 8364, 9170, 9508, 10366

Offset

8,2

Comments

In a semimagic square the row and column sums must all be equal (the "magic sum"). A "reduced" square has least entry 0. There is one normalized square for each symmetry class of reduced squares (symmetry under permutation of rows and columns and reflection in a diagonal).

a(n) is given by a quasipolynomial of degree 3 and period 60. [corrected by Hoang Xuan Thanh, Jul 16 2026]

References

Matthias Beck and Thomas Zaslavsky, An enumerative geometry for magic and magilatin labellings, Annals of Combinatorics, 10 (2006), no. 4, pages 395-413. MR 2007m:05010. Zbl 1116.05071.

Links

Thomas Zaslavsky, Table of n, a(n) for n = 8..10000.

Matthias Beck and Thomas Zaslavsky, Six Little Squares and How Their Numbers Grow , J. Int. Seq. 13 (2010), 10.6.2.

Matthias Beck and Thomas Zaslavsky, "Six Little Squares and How their Numbers Grow" Web Site: Maple worksheets and supporting documentation.

Index entries for linear recurrences with constant coefficients, signature (-2,-1,2,5,5,2,-3,-7,-7,-3,2,5,5,2,-1,-2,-1).

Formula

G.f.: x^5/((1-x)^3*(1-x^2)) - 2*x^5/((1-x)*(1-x^2)^2) - x^5/((1-x)^2*(1-x^3)) - 2*x^6/((1-x)*(1-x^2)*(1-x^3)) - x^6/(1-x^2)^3 - x^7/((1-x^2)^2*(1-x^3)) + x^5/((1-x)*(1-x^4)) + 2*x^5/((1-x^2)*(1-x^3)) + 2*x^6/((1-x^2)*(1-x^4)) + x^6/(1-x^3)^2 + x^7/((1-x^2)*(1-x^5)) + x^7/((1-x^3)*(1-x^4)) + x^8/((1-x^3)*(1-x^5)) - x^5/(1-x^5). - Thomas Zaslavsky, Mar 03 2010

a(n) ~ n^3/12. - Charles R Greathouse IV, May 20 2026

a(n) = (2*n^3-33*n^2+174*n-264)/24 + (n mod 2)*(n^2-16*n+50)/8 + ((2*n^2+1) mod 3)*(n-9)/3 + (n mod 3)*2/3 - 2*[(n mod 4)=0] - 2*[(n mod 5)=0]. - Hoang Xuan Thanh, Jul 16 2026

Example

For n=8 the cells contain 0,...,8, which have one semimagic arrangement up to symmetry. For n=9 the cells contain all of 0,...,9 except 3 or 6, since 0 and 9 must be used; each selection has one semimagic arrangement up to symmetry.

Prog

(PARI) Vec((18*x^19 + 10*x^18 - 5*x^17 - 18*x^16 - 14*x^15 - 10*x^14 - x^13 + 5*x^12 + 8*x^11 + 4*x^10 + 2*x^9 + x^8)/((1-x^2)^2*(1-x^3)^2*(1-x^4)*(1-x^5)) + O(x^60)) \\ Hoang Xuan Thanh, Jul 16 2026

Crossrefs

Cf. A173546, A173723. A173726 counts symmetry types by magic sum.

Sequence in context: A014775 A009218 A071122 * A101572 A080766 A262506

Adjacent sequences: A173721 A173722 A173723 * A173725 A173726 A173727

Keyword

nonn,easy,changed

Author

Thomas Zaslavsky, Feb 22 2010, Mar 03 2010

Status

approved