A173547 -id:A173547 - cp's OEIS frontend

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

Thomas Zaslavsky, Feb 21 2010

nonn

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

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.

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).

A173547 counts the same squares by magic sum.

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

G.f.: 72 * x^2/(1-x)^2 * { x^5/[(1-x)^3*(1-x^2)] - 2x^5/[(1-x)*(1-x^2)^2] - x^5/[(1-x)^2*(1-x^3)] - 2x^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)] + 2x^5/[(1-x^2)*(1-x^3)] + 2x^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

A173725 Number of symmetry classes of 3 X 3 semimagic squares with distinct positive values and magic sum n.

Original entry on oeis.org

1, 2, 4, 8, 12, 20, 29, 42, 54, 82, 97, 131, 169, 207, 249, 331, 372, 459, 551, 647, 745, 911, 1007, 1184, 1374, 1553, 1739, 2049, 2231, 2539, 2867, 3183, 3509, 3999, 4316, 4820, 5340, 5835, 6350, 7104, 7607, 8352, 9132, 9882, 10651, 11724, 12472, 13551

Offset: 15

Thomas Zaslavsky, Feb 23 2010

nonn

In a semimagic square the row and column sums must all be equal to the magic sum. The symmetries are permutation of rows and columns and reflection in a diagonal. a(n) is given by a quasipolynomial of degree 4 and period 840.

			a(15) is the first term because the values 1,...,9 make magic sum 15. By symmetries one can assume a_{11} is smallest, and a_{11} < a_{12} < a_{21} < a_{31} < a_{13}. a(15)=1 because there is only one normal form with values 1,...,9 (equivalent to the classical 3 X 3 magic square). a(16)=2 because the values 1,...,8,10 give two normal forms.

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.

Thomas Zaslavsky, Table of n, a(n) for n = 15..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, -3, -2, 0, 3, 6, 8, 9, 7, 3, -4, -10, -15, -16, -14, -8, 0, 8, 14, 16, 15, 10, 4, -3, -7, -9, -8, -6, -3, 0, 2, 3, 2, 1).

Cf. A173547, A173726. A173723 counts symmetry types by largest cell value.

G.f.: (x^3)/(1-x^3) * { x^7/[(x-1)*(x^2-1)^3] + 2x^7/[(x-1)*(x^2-1)*(x^4-1)] + x^7/[(x-1)*(x^6-1)] + x^7/[(x^2-1)^2*(x^3-1)] + x^7/[(x^2-1)*(x^5-1)] + x^7/[(x^3-1)*(x^4-1)] + x^7/(x^7-1) + x^9/[(x-1)*(x^4-1)^2] + 2*x^9/[(x^2-1)*(x^3-1)*(x^4-1)] + 2*x^9/[(x^3-1)*(x^6-1)] + x^9/[(x^4-1)*(x^5-1)] + x^11/[(x^3-1)*(x^4-1)^2] + x^11/[(x^3-1)*(x^8-1)] + x^11/[(x^5-1)*(x^6-1)] + x^13/[(x^5-1)*(x^8-1)] }. - Thomas Zaslavsky, Mar 03 2010

A173726 Number of reduced, normalized 3x3 semimagic squares with magic sum n.

Original entry on oeis.org

1, 2, 4, 7, 10, 16, 21, 30, 34, 53, 55, 77, 87, 110, 118, 162, 165, 210, 220, 275, 286, 360, 360, 439, 463, 546, 555, 675, 678, 800, 818, 952, 970, 1132, 1133, 1311, 1341, 1519, 1530, 1764, 1772, 2002, 2028, 2275, 2299, 2592, 2590, 2900, 2939, 3250, 3265, 3644

Offset: 12

Thomas Zaslavsky, Feb 23 2010

nonn

In a semimagic square the row and column sums must all equal the magic sum. The symmetries are permutation of rows and columns and reflection in a diagonal. A "reduced" square has least entry 0. There is one normalized square for each symmetry class of reduced squares. See A173725 for a general normal form. a(n) is given by a quasipolynomial of degree 4 and period 840.

			a(12) is the first term because the values 0,...,8 make magic sum 12. a(12)=1 because there is only one normal form with values 0 to 8: (by rows) 0,4,8;5,6,1;7,2,3. a(13)=2 because the values 0,...,5,7,8,9 give two normal forms: 0,4,9;5,7,1;8,2,3 and 0,4,9;5,7,1;8,2,3.

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.

Thomas Zaslavsky, Table of n, a(n) for n=12..10000.
Matthias Beck, 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, -3, -3, -2, 0, 3, 6, 9, 10, 9, 5, 0, -6, -11, -14, -14, -11, -6, 0, 5, 9, 10, 9, 6, 3, 0, -2, -3, -3, -2, -1).

Cf. A173547, A173725. A173724 counts squares by largest cell value.

A173728 Number of reduced 3 X 3 semimagic squares with magic sum n.

Original entry on oeis.org

72, 144, 288, 504, 720, 1152, 1512, 2160, 2448, 3816, 3960, 5544, 6264, 7920, 8496, 11664, 11880, 15120, 15840, 19800, 20592, 25920, 25920, 31608, 33336, 39312, 39960, 48600, 48816, 57600, 58896, 68544, 69840, 81504, 81576, 94392, 96552

Offset: 12

Thomas Zaslavsky, Mar 03 2010

nonn

In a semimagic square the row and column sums must all equal the magic sum. The symmetries are permutation of rows and columns and reflection in a diagonal. A "reduced" square has least entry 0.

a(n) is given by a quasipolynomial of degree 4 and period 840.

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.

Thomas Zaslavsky, Table of n, a(n) for n = 12..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, -3, -3, -2, 0, 3, 6, 9, 10, 9, 5, 0, -6, -11, -14, -14, -11, -6, 0, 5, 9, 10, 9, 6, 3, 0, -2, -3, -3, -2, -1).

Cf. A173547, A173725, A173726. A173724 counts squares by largest cell value.

G.f.: 72 * { x^7/[(x-1)*(x^2-1)^3] + 2x^7/[(x-1)*(x^2-1)*(x^4-1)] + x^7/[(x-1)*(x^6-1)] + x^7/[(x^2-1)^2*(x^3-1)] + x^7/[(x^2-1)*(x^5-1)] + x^7/[(x^3-1)*(x^4-1)] + x^7/(x^7-1) + x^9/[(x-1)*(x^4-1)^2] + 2*x^9/[(x^2-1)*(x^3-1)*(x^4-1)] + 2*x^9/[(x^3-1)*(x^6-1)] + x^9/[(x^4-1)*(x^5-1)] + x^11/[(x^3-1)*(x^4-1)^2] + x^11/[(x^3-1)*(x^8-1)] + x^11/[(x^5-1)*(x^6-1)] + x^13/[(x^5-1)*(x^8-1)] }.

cp's OEIS Frontend

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").

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A173725 Number of symmetry classes of 3 X 3 semimagic squares with distinct positive values and magic sum n.

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A173726 Number of reduced, normalized 3x3 semimagic squares with magic sum n.

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A173728 Number of reduced 3 X 3 semimagic squares with magic sum n.

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