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A174256
Number of 3 X 3 reduced magic squares with distinct values and maximum value 2n; also, with magic sum 3n.
2
0, 0, 0, 8, 16, 8, 24, 24, 24, 32, 40, 32, 48, 48, 48, 56, 64, 56, 72, 72, 72, 80, 88, 80, 96, 96, 96, 104, 112, 104, 120, 120, 120, 128, 136, 128, 144, 144, 144, 152, 160, 152, 168, 168, 168, 176, 184, 176, 192, 192, 192, 200, 208, 200, 216, 216, 216, 224, 232, 224
OFFSET
1,4
COMMENTS
In a reduced magic square the row, column, and two diagonal sums must all be equal (the "magic sum") and the minimum entry is 0. The maximum entry is necessarily even and = (2/3)*(magic sum).
a(n) is a quasipolynomial with period 6.
The second differences of A108576 are a(n/2) for even n and 0 for odd n. The first differences of A108578 are a(n).
LINKS
Matthias Beck and Thomas Zaslavsky, Six Little Squares and How Their Numbers Grow , J. Int. Seq. 13 (2010), 10.6.2.
Matthias Beck and Thomas Zaslavsky, "Six Little Squares and How their Numbers Grow" Web Site: Maple worksheets and supporting documentation.
FORMULA
a(n) = 8*A174257(n).
G.f.: 8*x^4 * (2*x+1) / ((x^2-1) * (x^3-1)). [amended by Georg Fischer, Apr 17 2020]
a(n) = 2*(6*n - 13 - 8*cos(2*n*Pi/3) - 3*cos(n*Pi))/3. - Wesley Ivan Hurt, Oct 04 2018
MATHEMATICA
Take[CoefficientList[Series[(8x^8 (2x^2+1))/((x^4-1)(x^6-1)), {x, 0, 120}], x], {1, -1, 2}] (* Harvey P. Dale, Aug 07 2017 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Thomas Zaslavsky, Mar 14 2010
STATUS
approved