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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
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).
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-François Alcover, Nov 17 2018 *)
CROSSREFS
Cf. A173549 (all squares), A173548 (counted by upper bound), A173729 (symmetry types by upper bound).
Sequence in context: A217385 A138827 A353055 * A036685 A034744 A219748
KEYWORD
nonn
AUTHOR
Thomas Zaslavsky, Mar 04 2010, Apr 24 2010
STATUS
approved

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Last modified March 18 22:48 EDT 2024. Contains 370951 sequences. (Running on oeis4.)