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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

%I

%S 0,0,0,8,16,8,24,24,24,32,40,32,48,48,48,56,64,56,72,72,72,80,88,80,

%T 96,96,96,104,112,104,120,120,120,128,136,128,144,144,144,152,160,152,

%U 168,168,168,176,184,176,192,192,192,200,208,200,216,216,216,224,232,224

%N Number of 3 X 3 reduced magic squares with distinct values and maximum value 2n; also, with magic sum 3n.

%C 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).

%C a(n) = 8*A174257.

%C a(n) is a quasipolynomial with period 6.

%C 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).

%H T. Zaslavsky, <a href="/A174256/b174256.txt">Table of n, a(n) for n = 1..10000</a>.

%H M. Beck, T. Zaslavsky, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Zaslavsky/sls.html">Six Little Squares and How Their Numbers Grow </a>, J. Int. Seq. 13 (2010), 10.6.2.

%H Matthias Beck and Thomas Zaslavsky, <a href="http://www.math.binghamton.edu/zaslav/Tmath/SLSfiles/">"Six Little Squares and How their Numbers Grow" Web Site</a>: Maple worksheets and supporting documentation.

%F G.f.: 8*x^4 * (2*x+1) / ((x^2-1) * (x^3-1)). [amended by _Georg Fischer_, Apr 17 2020]

%F a(n) = (1/45)*(4*(n mod 6) + 4*((n+1) mod 6) - 116*((n+2) mod 6) - 116*((n+3) mod 6) - 56*((n+4) mod 6) - 56*((n+5) mod 6)) + (4/15)*Sum_{k=0..n} (-14*(k mod 6) + ((k+1) mod 6) + ((k+2) mod 6) + ((k+3) mod 6) + ((k+4) mod 6) + 16*((k+5) mod 6)), with n >= 0. - _Paolo P. Lava_ and _Giorgio Balzarotti_, Mar 22 2010

%F a(n) = 2*(6*n - 13 - 8*cos(2*n*Pi/3) - 3*cos(n*Pi))/3. - _Wesley Ivan Hurt_, Oct 04 2018

%t 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 *)

%Y Cf. A108576, A108577, A174257.

%K nonn,easy

%O 1,4

%A _Thomas Zaslavsky_, Mar 14 2010

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Last modified June 25 04:15 EDT 2021. Contains 345452 sequences. (Running on oeis4.)