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A108578 Number of 3 X 3 magic squares with magic sum 3n. 7
0, 0, 0, 0, 8, 24, 32, 56, 80, 104, 136, 176, 208, 256, 304, 352, 408, 472, 528, 600, 672, 744, 824, 912, 992, 1088, 1184, 1280, 1384, 1496, 1600, 1720, 1840, 1960, 2088, 2224, 2352, 2496, 2640, 2784, 2936, 3096, 3248, 3416, 3584, 3752, 3928, 4112, 4288 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
Contribution from Thomas Zaslavsky, Mar 12 2010: (Start)
A magic square has distinct positive integers in its cells, whose sum is the same (the "magic sum") along any row, column, or main diagonal.
a(n) is given by a quasipolynomial of period 6. (End)
LINKS
M. Beck and T. Zaslavsky, An enumerative geometry for magic and magilatin labellings, Ann. Combinatorics, 10 (2006), no. 4, 395-413. MR 2007m:05010. Zbl 1116.05071. - Thomas Zaslavsky, Jan 29 2010
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.: [8*x^5*(1+2*x)] / [(1-x)*(1-x^2)*(1-x^3)].
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6). - Vincenzo Librandi, Sep 01 2018
EXAMPLE
a(5) = 8 because there are 8 3 X 3 magic squares with entries 1,...,9 and magic sum 15.
MATHEMATICA
LinearRecurrence[{1, 1, 0, -1, -1, 1}, {0, 0, 0, 0, 8, 24}, 50] (* Jean-François Alcover, Sep 01 2018 *)
CoefficientList[Series[8 x^4 (1 + 2 x) / ((1 - x) (1 - x^2) (1 - x^3)), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 01 2018 *)
PROG
(PARI) a(n)=(1/9)*(2*n^2-32*n+[144, 78, 120, 126, 96, 102][(n%18)/3+1])
(PARI) x='x+O('x^99); concat(vector(4), Vec(8*x^5*(1+2*x)/((1-x)*(1-x^2)*(1-x^3)))) \\ Altug Alkan, Sep 01 2018
(Magma) I:=[0, 0, 0, 0, 8, 24]; [n le 6 select I[n] else Self(n-1)+Self(n-2)-Self(n-4)-Self(n-5)+Self(n-6): n in [1..60]]; // Vincenzo Librandi, Sep 01 2018
CROSSREFS
Equals 8 times the second differences of A055328.
Sequence in context: A283078 A319528 A140403 * A305241 A044450 A134223
KEYWORD
nonn
AUTHOR
EXTENSIONS
Edited by N. J. A. Sloane, Feb 05 2010
Corrected g.f. to account for previous change in parameter n from magic sum s to s/3; by Thomas Zaslavsky, Mar 12 2010
STATUS
approved

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Last modified July 14 05:06 EDT 2024. Contains 374291 sequences. (Running on oeis4.)