A173730 -id:A173730 - cp's OEIS frontend

A173548 Number of 3 X 3 magilatin squares with positive values < n.

12, 48, 120, 384, 1068, 2472, 4896, 9072, 15516, 25608, 40296, 61608, 91068, 131640, 185136, 255960, 346860, 463248, 608088, 789240, 1010316, 1280544, 1604832, 1994064, 2454012, 2998656, 3633912, 4376064, 5232972, 6223080, 7354896

Offset: 4

Thomas Zaslavsky, Mar 03 2010, Apr 24 2010

nonn

A magilatin squares has equal row and column sums and no number repeated in any row or column.

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

T. Zaslavsky, Table of n, a(n) for n=4..10000.
Matthias Beck and Thomas Zaslavsky, An enumerative geometry for magic and magilatin labellings, arXiv:math/0506315 [math.CO], 2005.
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.
Matthias Beck and Thomas Zaslavsky, Six little squares and how their numbers grow, Journal of Integer Sequences, 13 (2010), Article 10.6.2.
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).

Cf. A173729 (symmetry types), A173549 (counted by magic sum), A173730 (symmetry types by magic sum).

LinearRecurrence[{0, 2, 2, 0, -3, -3, -2, 1, 4, 4, 1, -2, -3, -3, 0, 2, 2, 0, -1}, {12, 48, 120, 384, 1068, 2472, 4896, 9072, 15516, 25608, 40296, 61608, 91068, 131640, 185136, 255960, 346860, 463248, 608088}, 31] (* Jean-François Alcover, Nov 05 2018 *)

G.f.: x^2/(1-x)^2 * { 12x^2/(x-1)^2 - 36x^3/(x-1)^3 - 72x^3/[(x-1)*(x^2-1)] - 36x^3/(x^3-1) - 72x^4/[(x-1)^2*(x^2-1)] - 36x^4/[(x-1)*(x^3-1)] - 72x^4/(x^2-1)^2 + 72x^5/[(x-1)^3*(x^2-1)] + 72x^5/[(x-1)^2*(x^3-1)] + 144x^5/[(x-1)*(x^2-1)^2] + 72x^5/[(x-1)*(x^4-1)] + 108x^5/[(x^2-1)*(x^3-1)] + 72x^5/(x^5-1) + 144x^6/[(x-1)*(x^2-1)*(x^3-1)] + 72x^6/(x^2-1)^3 + 144x^6/[(x^2-1)*(x^4-1)] + 72x^6/(x^3-1)^2 + 72x^7/[(x^2-1)^2*(x^3-1)] + 72x^7/[(x^2-1)*(x^5-1)] + 72x^7/[(x^3-1)*(x^4-1)] + 72x^8/[(x^3-1)*(x^5-1)] }.

A173549 Number of 3 X 3 magilatin squares with positive values and magic sum n.

Original entry on oeis.org

12, 12, 24, 72, 156, 240, 552, 600, 1020, 1548, 2004, 2568, 4008, 4644, 6264, 8136, 10152, 12168, 16284, 18372, 22992, 27972, 32736, 37896, 47352, 52332, 62004, 72288, 82572, 93108, 110280, 120492, 138420, 157428, 175248, 193824, 223428

Offset: 6

Thomas Zaslavsky, Mar 04 2010, Apr 24 2010

nonn

A magilatin square has equal row and column sums and no number repeated in any row or column.

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

Ray Chandler, Table of n, a(n) for n = 6..10000
Matthias Beck and Thomas Zaslavsky, An enumerative geometry for magic and magilatin labellings, arXiv:math/0506315 [math.CO], 2005
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.
Matthias Beck and Thomas Zaslavsky, Six Little Squares and How their Numbers Grow, Journal of Integer Sequences, 13 (2010), Article 10.6.2.
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. A173730 (symmetry types), A173548 (counted by upper bound), A173729 (symmetry types by upper bound).

LinearRecurrence[{-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}, {0, 0, 0, 0, 0, 12, 12, 24, 72, 156, 240, 552, 600, 1020, 1548, 2004, 2568, 4008, 4644, 6264, 8136, 10152, 12168, 16284, 18372, 22992, 27972, 32736, 37896, 47352, 52332, 62004, 72288, 82572}, 42][[6;;]] (* Jean-François Alcover, Nov 06 2018 *)

G.f.: x^3/(1-x^3) * { 12*x^3/[(x-1)*(x^2-1)] - 108*x^5/[(x-1)*(x^2-1)^2] - 72*x^5/[(x-1)*(x^4-1)] - 72*x^5/[(x^3-1)*(x^2-1)] - 36*x^5/(x^5-1) + 72*x^7/[(x-1)*(x^2-1)^3] + 144*x^7/[(x-1)*(x^2-1)*(x^4-1)] + 72*x^7/[(x-1)*(x^6-1)] + 72*x^7/[(x^2-1)^2*(x^3-1)] + 72*x^7/[(x^2-1)*(x^5-1)] + 72*x^7/(x^7-1) + 72*x^9/[(x-1)*(x^4-1)^2] + 144*x^9/[(x^2-1)*(x^3-1)*(x^4-1)] + 144*x^9/[(x^3-1)*(x^6-1)] + 72*x^9/[(x^4-1)*(x^5-1)] + 72*x^11/[(x^3-1)*(x^4-1)^2] + 72*x^11/[(x^3-1)*(x^8-1)] + 72*x^11/[(x^5-1)*(x^6-1)] + 72*x^13/[(x^5-1)*(x^8-1)] }.

A173729 Number of symmetry classes of 3 X 3 magilatin squares with positive values < n.

Original entry on oeis.org

1, 4, 10, 24, 53, 106, 191, 328, 528, 822, 1230, 1794, 2542, 3534, 4802, 6428, 8460, 10996, 14087, 17870, 22405, 27850, 34286, 41896, 50773, 61148, 73116, 86942, 102751, 120840, 141343, 164618, 190808, 220306, 253292, 290202, 331226, 376872

Offset: 4

Thomas Zaslavsky, Mar 04 2010, Apr 24 2010

A magilatin square has equal row and column sums and no number repeated in any row or column. The symmetries are row and column permutations and diagonal flip.

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

Thomas Zaslavsky, Table of n, a(n) for n = 4..10000.
Matthias Beck and Thomas Zaslavsky, An enumerative geometry for magic and magilatin labellings, Ann. Combinatorics, 10 (2006), no. 4, 395-413. MR 2007m:05010. Zbl 1116.05071.
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).

Cf. A173548 (total number of squares), A173549 (squares counted by magic sum), A173730 (symmetry types by magic sum).

CoefficientList[Series[x^4*(1 + 4*x + 8*x^2 + 14*x^3 + 25*x^4 + 41*x^5 + 52*x^6 + 54*x^7 + 43*x^8 + 27*x^9 + 13*x^10 + 10*x^11 + 16*x^12 + 23*x^13 + 20*x^14 + 9*x^15)/((1 + x^2)*(1 + x)^3*(1 + x + x^2)^2*(1 + x + x^2 + x^3 + x^4)*(1 - x)^6), {x, 0, 41}], x] (* L. Edson Jeffery, Sep 10 2017 *)

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

G.f.: x^4*(1 + 4*x + 8*x^2 + 14*x^3 + 25*x^4 + 41*x^5 + 52*x^6 + 54*x^7 + 43*x^8 + 27*x^9 + 13*x^10 + 10*x^11 + 16*x^12 + 23*x^13 + 20*x^14 + 9*x^15)/((1 + x^2)*(1 + x)^3*(1 + x + x^2)^2*(1 + x + x^2 + x^3 + x^4)*(1 - x)^6). - L. Edson Jeffery, Sep 10 2017

A174021 Number of symmetry classes of reduced 3x3 magilatin squares with magic sum n.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 16, 15, 25, 30, 41, 43, 66, 68, 92, 99, 129, 136, 180, 180, 231, 245, 297, 304, 385, 388, 469, 482, 575, 588, 706, 704, 831, 858, 987, 996, 1171, 1175, 1350, 1370, 1561, 1581, 1806, 1804, 2047, 2081, 2323, 2335, 2641, 2649, 2951, 2979, 3302

Offset: 3

Thomas Zaslavsky, Mar 05 2010

nonn

A magilatin square has equal row and column sums and no number repeated in any row or column. It is reduced if the least value in it is 0. The symmetries are row and column permutations and diagonal flip.

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=3..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. A173549 (all magilatin squares), A173730 (symmetry types), A174020 (reduced squares), A174019 (reduced symmetry types by largest value).

"Distinct" values (incorrect) deleted by Thomas Zaslavsky, Apr 24 2010

A174018 Number of reduced 3 X 3 magilatin squares with largest entry n.

Original entry on oeis.org

12, 24, 36, 192, 420, 720, 1020, 1752, 2268, 3648, 4596, 6624, 8148, 11112, 12924, 17328, 20076, 25488, 28452, 36312, 39924, 49152, 54060, 64944, 70716, 84696, 90612, 106896, 114756, 133200, 141708, 164184, 173340, 198192, 209796, 237600

Offset: 2

Thomas Zaslavsky, Mar 05 2010

nonn

A magilatin square has equal row and column sums and no number repeated in any row or column. It is reduced if the least value in it is 0.

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 = 2..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).

Cf. A173548 (all magilatin squares), A173730 (symmetry types), A174019 (reduced symmetry types by largest value), A174020 (reduced squares by magic sum), A174021 (reduced symmetry types by magic sum).

G.f.: 12x^2/(x-1)^2 - 36x^3/(x-1)^3 - 72x^3/[(x-1)*(x^2-1)] - 36x^3/(x^3-1) - 72x^4/[(x-1)^2*(x^2-1)] - 36x^4/[(x-1)*(x^3-1)] - 72x^4/(x^2-1)^2 + 72x^5/[(x-1)^3*(x^2-1)] + 72x^5/[(x-1)^2*(x^3-1)] + 144x^5/[(x-1)*(x^2-1)^2] + 72x^5/[(x-1)*(x^4-1)] + 108x^5/[(x^2-1)*(x^3-1)] + 72x^5/(x^5-1) + 144x^6/[(x-1)*(x^2-1)*(x^3-1)] + 72x^6/(x^2-1)^3 + 144x^6/[(x^2-1)*(x^4-1)] + 72x^6/(x^3-1)^2 + 72x^7/[(x^2-1)^2*(x^3-1)] + 72x^7/[(x^2-1)*(x^5-1)] + 72x^7/[(x^3-1)*(x^4-1)] + 72x^8/[(x^3-1)*(x^5-1)].

"Distinct" values (incorrect) deleted by Thomas Zaslavsky, Apr 24 2010

A174019 Number of symmetry classes of reduced 3 X 3 magilatin squares with largest entry n.

Original entry on oeis.org

1, 2, 3, 8, 15, 24, 32, 52, 63, 94, 114, 156, 184, 244, 276, 358, 406, 504, 555, 692, 752, 910, 991, 1174, 1267, 1498, 1593, 1858, 1983, 2280, 2414, 2772, 2915, 3308, 3488, 3924, 4114, 4622, 4816, 5374, 5616, 6216, 6467, 7154, 7418, 8158, 8469, 9264, 9587

Offset: 2

Thomas Zaslavsky, Mar 05 2010

nonn

A magilatin square has equal row and column sums and no number repeated in any row or column. It is reduced if the least value in it is 0. The symmetries are row and column permutations and diagonal flip.

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 = 2..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).

Cf. A173548 (all magilatin squares), A173730 (symmetry types), A174018 (reduced squares by largest value), A174021 (reduced symmetry types by magic sum).

G.f.: x^2/(x-1)^2 - x^3/(x-1)^3 - 2x^3/[(x-1)*(x^2-1)] - x^3/(x^3-1) - 2x^4/[(x-1)^2*(x^2-1)] - x^4/[(x-1)*(x^3-1)] - 2x^4/(x^2-1)^2 + x^5/[(x-1)^3*(x^2-1)] + x^5/[(x-1)^2*(x^3-1)] + 2x^5/[(x-1)*(x^2-1)^2] + x^5/[(x-1)*(x^4-1)] + x^5/[(x^2-1)*(x^3-1)] + x^5/(x^5-1) + 2x^6/[(x-1)*(x^2-1)*(x^3-1)] + 2x^6/[(x^2-1)*(x^4-1)] + x^6/(x^2-1)^3 + x^6/(x^3-1)^2 + x^7/[(x^3-1)*(x^4-1)] + x^7/[(x^2-1)*(x^5-1)] + x^7/[(x^2-1)^2*(x^3-1)] + x^8/[(x^3-1)*(x^5-1)]

"Distinct" values (incorrect) deleted by Thomas Zaslavsky, Apr 24 2010

A174020 Number of reduced 3 X 3 magilatin squares with magic sum n.

Original entry on oeis.org

12, 12, 24, 60, 144, 216, 480, 444, 780, 996, 1404, 1548, 2460, 2640, 3696, 4128, 5508, 5904, 8148, 8220, 10824, 11688, 14364, 14904, 19380, 19596, 24108, 24936, 30240, 31104, 37992, 37920, 45312, 47148, 54756, 55404, 66000, 66252, 76920, 78288

Offset: 3

Thomas Zaslavsky, Mar 05 2010

nonn

A magilatin square has equal row and column sums and no number repeated in any row or column. It is reduced if the least value in it is 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 = 3..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. A173549 (all magilatin squares), A173730 (symmetry types), A174021 (reduced symmetry types), A174018 (reduced squares by largest value).

G.f.: 12*x^3/[(x-1)*(x^2-1)] - 108*x^5/[(x-1)*(x^2-1)^2] - 72*x^5/[(x-1)*(x^4-1)] - 72*x^5/[(x^3-1)*(x^2-1)] - 36*x^5/(x^5-1) + 72*x^7/[(x-1)*(x^2-1)^3] + 144*x^7/[(x-1)*(x^2-1)*(x^4-1)] + 72*x^7/[(x-1)*(x^6-1)] + 72*x^7/[(x^2-1)^2*(x^3-1)] + 72*x^7/[(x^2-1)*(x^5-1)] + 72*x^7/(x^7-1) + 72*x^9/[(x-1)*(x^4-1)^2] + 144*x^9/[(x^2-1)*(x^3-1)*(x^4-1)] + 144*x^9/[(x^3-1)*(x^6-1)] + 72*x^9/[(x^4-1)*(x^5-1)] + 72*x^11/[(x^3-1)*(x^4-1)^2] + 72*x^11/[(x^3-1)*(x^8-1)] + 72*x^11/[(x^5-1)*(x^6-1)] + 72*x^13/[(x^5-1)*(x^8-1)].

"Distinct" values (incorrect) deleted by Thomas Zaslavsky, Apr 24 2010

cp's OEIS Frontend

A173548 Number of 3 X 3 magilatin squares with positive values < n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A173549 Number of 3 X 3 magilatin squares with positive values and magic sum n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A173729 Number of symmetry classes of 3 X 3 magilatin squares with positive values < n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A174021 Number of symmetry classes of reduced 3x3 magilatin squares with magic sum n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A174018 Number of reduced 3 X 3 magilatin squares with largest entry n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

Extensions

A174019 Number of symmetry classes of reduced 3 X 3 magilatin squares with largest entry n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

Extensions

A174020 Number of reduced 3 X 3 magilatin squares with magic sum n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

Extensions