cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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

Views

Author

Thomas Zaslavsky, Mar 04 2010, Apr 24 2010

Keywords

Comments

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.

Links

Crossrefs

Cf. A173730 (symmetry types), A173548 (counted by upper bound), A173729 (symmetry types by upper bound).

Programs

  • Mathematica
    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 *)

Formula

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)] }.

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

Views

Author

Thomas Zaslavsky, Mar 04 2010, Apr 24 2010

Keywords

Comments

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.

Links

Crossrefs

Cf. A173549 (all squares), A173548 (counted by upper bound), A173729 (symmetry types by upper bound).

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

Thomas Zaslavsky, Feb 21 2010

Keywords

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

Crossrefs

A173547 counts the same squares by magic sum.
Cf. A108576, A108577, A108578, A108579, A173548, A173549.

Formula

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

Views

Author

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

Keywords

Comments

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]

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

Crossrefs

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

Formula

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

Views

Author

Thomas Zaslavsky, Mar 04 2010, Apr 24 2010

Keywords

Comments

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.

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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

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

Views

Author

Thomas Zaslavsky, Mar 05 2010

Keywords

Comments

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.

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

Crossrefs

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).

Formula

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)].

Extensions

"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

Views

Author

Thomas Zaslavsky, Mar 05 2010

Keywords

Comments

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.

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

Crossrefs

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

Formula

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)]

Extensions

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