A173546 -id:A173546 - cp's OEIS frontend

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

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)] }

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

Original entry on oeis.org

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

Thomas Zaslavsky, Feb 22 2010, Mar 03 2010

nonn

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 5 and period 60.

			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.

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 = 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).

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

G.f.: 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

A173723 Number of symmetry classes of 3 X 3 semimagic squares with distinct positive values < n.

Original entry on oeis.org

1, 4, 13, 36, 80, 160, 291, 496, 794, 1226, 1821, 2632, 3691, 5080, 6840, 9070, 11826, 15228, 19344, 24332, 30262, 37322, 45606, 55330, 66597, 79674, 94673, 111892, 131474, 153756, 178891, 207278, 239074, 274724, 314427, 358666, 407649, 461936

Offset: 10

Thomas Zaslavsky, Feb 22 2010, Mar 03 2010

In a semimagic square the row and column sums must all be equal (the "magic sum"). Symmetry is up to permutation of rows and columns and reflection in a diagonal. a(n) is given by a quasipolynomial of degree 5 and period 60.

			For a(10) the cells contain the nine integers from 1 to 9, which can be arranged in 1 way to make a magic square, up to symmetry. For a(11) the cells contain nine of the ten integers from 1 to 10. The omitted number can only be 1, 4, 7, or 10. Each selection of numbers can be arranged in 1 way, up to symmetry.

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).

Cf. A173546, A173724. A173725 counts symmetry types by magic sum.

G.f.: 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

A173727 Number of reduced 3 X 3 semimagic squares with distinct nonnegative integer entries and maximum entry n.

Original entry on oeis.org

72, 144, 432, 1008, 1512, 2592, 3672, 5328, 6696, 9648, 11736, 15552, 17856, 23760, 26712, 33840, 37872, 46512, 51408, 62784, 67824, 81360, 88128, 103680, 111096, 130320, 138384, 159840, 170136, 194400, 205416, 234144, 245448, 277488, 291816

Offset: 8

Thomas Zaslavsky, Mar 03 2010

In a semimagic square the row and column sums must all be equal (the "magic sum"). A reduced square has least entry 0.

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

			For n=8 the cells contain 0,...,8, which have one semimagic arrangement up to symmetry. All examples are obtained by symmetries from (by rows): 0, 5, 7; 4, 6, 2; 8, 1, 3.
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.

Thomas Zaslavsky, Table of n, a(n) for n = 8..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 , 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. A173546, A173723, A173724. A173728 counts reduced squares by magic sum.

G.f.: 72 * ( x^5/((x-1)*(x^4-1)) + x^5/((x-1)^2*(x^3-1)) + x^5/((x-1)^3*(x^2-1)) + 2*x^5/((x-1)*(x^2-1)^2) + 2*x^5/((x^2-1)*(x^3-1)) + x^5/(x^5-1) + 2*x^6/((x-1)*(x^2-1)*(x^3-1)) + x^6/(x^2-1)^3 + 2*x^6/((x^2-1)*(x^4-1)) + x^6/(x^3-1)^2 + x^7/((x^2-1)*(x^5-1)) + x^7/((x^2-1)^2*(x^3-1)) + x^7/((x^3-1)*(x^4-1)) + x^8/((x^3-1)*(x^5-1)) ).

cp's OEIS Frontend

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

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

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A173723 Number of symmetry classes of 3 X 3 semimagic squares with distinct positive values < n.

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A173727 Number of reduced 3 X 3 semimagic squares with distinct nonnegative integer entries and maximum entry n.

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