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A173730 Number of symmetry classes of 3 X 3 magilatin squares with positive values and magic sum n. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

6,3

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

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.

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)) ).

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-Fran├žois Alcover, Nov 17 2018 *)

CROSSREFS

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

Sequence in context: A088762 A217385 A138827 * A036685 A034744 A219748

Adjacent sequences:  A173727 A173728 A173729 * A173731 A173732 A173733

KEYWORD

nonn

AUTHOR

Thomas Zaslavsky, Mar 04 2010, Apr 24 2010

STATUS

approved

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Last modified August 18 13:58 EDT 2019. Contains 326100 sequences. (Running on oeis4.)