A173546 Number of 3 X 3 semimagic squares with distinct positive values < n. In a semimagic squares the row and column sums must all be equal (the "magic sum").

72, 288, 936, 2592, 5760, 11520, 20952, 35712, 57168, 88272, 131112, 189504, 265752, 365760, 492480, 653040, 851472, 1096416, 1392768, 1751904, 2178864, 2687184, 3283632, 3983760, 4794984, 5736528, 6816456, 8056224, 9466128

Offset

10,1

Comments

a(n) is given by a quasipolynomial of degree 5 and period 60.

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 = 10..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 (0,2,2,0,-3,-3,-2,1,4,4,1,-2,-3,-3,0,2,2,0,-1).

Formula

G.f.: 72 * x^2/(1-x)^2 * ( 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) ~ 3*n^5/10. - Charles R Greathouse IV, May 20 2026

a(n) = 3/10*n^5 - 75/8*n^4 + 331/3*n^3 + O(n^2). - Charles R Greathouse IV, May 20 2026

a(n) = 3*n^5/10 - 75*n^4/8 + 331*n^3/3 - 5989*n^2/10 + 1464*n - 18526/15 + (n mod 2)*(18*n^2-324*n+1629)/8 - ((2*n^2+2*n) mod 3)*(24*n-280)/3 - ((2*n^2+1) mod 3)*64/3 + 18*(((n+1) mod 4) - (n mod 4)) - ((2*n^2+n+2) mod 5)*144/5. - Hoang Xuan Thanh, Jul 16 2026

Prog

(PARI) Vec(x^10*(1296*x^9 + 3312*x^8 + 4968*x^7 + 5328*x^6 + 4680*x^5 + 3312*x^4 + 1872*x^3 + 792*x^2 + 288*x^1 + 72)/((1-x^2)^2*(1-x^3)^2*(1-x^4)*(1-x^5)) + O(x^60)) \\ Hoang Xuan Thanh, Jul 16 2026

Crossrefs

A173547 counts the same squares by magic sum.

Cf. A108576, A108577, A108578, A108579, A173548, A173549.

Sequence in context: A004007 A279272 A378002 * A308136 A242534 A277430

Adjacent sequences: A173543 A173544 A173545 * A173547 A173548 A173549

Keyword

nonn,easy,changed

Author

Thomas Zaslavsky, Feb 21 2010

Status

approved