A173549 -id:A173549 - 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)] }.

A173730 Number of symmetry classes of 3 X 3 magilatin squares with positive values and magic sum n.

Original entry on oeis.org

1, 1, 2, 4, 7, 10, 20, 22, 35, 50, 63, 78, 116, 131, 170, 215, 260, 306, 395, 440, 537, 640, 737, 841, 1025, 1125, 1310, 1507, 1700, 1898, 2213, 2404, 2729, 3071, 3391, 3725, 4242, 4566, 5075, 5612, 6127, 6656, 7418, 7931, 8703, 9499, 10254, 11038, 12140

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. The symmetries are row and column permutations and diagonal flip.

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

T. Zaslavsky, 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. A173549 (all squares), 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}, {1, 1, 2, 4, 7, 10, 20, 22, 35, 50, 63, 78, 116, 131, 170, 215, 260, 306, 395, 440, 537, 640, 737, 841, 1025, 1125, 1310, 1507, 1700, 1898, 2213, 2404, 2729, 3071}, 50] (* Jean-François Alcover, Nov 17 2018 *)

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

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").

Original entry on oeis.org

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

A173547 Number of 3 X 3 semimagic squares with distinct positive values and magic sum n.

Original entry on oeis.org

72, 144, 288, 576, 864, 1440, 2088, 3024, 3888, 5904, 6984, 9432, 12168, 14904, 17928, 23832, 26784, 33048, 39672, 46584, 53640, 65592, 72504, 85248, 98928, 111816, 125208, 147528, 160632, 182808, 206424, 229176, 252648, 287928, 310752

Offset: 15

Thomas Zaslavsky, Feb 21 2010, Feb 24 2010, Mar 03 2010

nonn

In a semimagic squares the row and column sums must all be equal to the magic sum. 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. [From Thomas Zaslavsky, Mar 03 2010]

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).

A173546 counts the same squares by upper bound on the entries. Cf. A108576, A108577, A108578, A108579, A173548, A173549.

G.f.: 72 * 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)] }

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

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

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

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A173730 Number of symmetry classes of 3 X 3 magilatin squares with positive values and magic sum n.

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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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A173547 Number of 3 X 3 semimagic squares with distinct positive values and magic sum n.

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

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

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

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