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.

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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

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Author

Thomas Zaslavsky, Feb 22 2010, Mar 03 2010

Keywords

Comments

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.

Examples

			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.

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

Formula

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

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

Views

Author

Thomas Zaslavsky, Feb 23 2010

Keywords

Comments

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.

Examples

			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.

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

Formula

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

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

Views

Author

Thomas Zaslavsky, Mar 03 2010

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Cf. A173546, A173723, A173724. A173728 counts reduced squares by magic sum.

Formula

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)) ).
Showing 1-3 of 3 results.

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