A174256 Number of 3 X 3 reduced magic squares with distinct values and maximum value 2n; also, with magic sum 3n.

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

Author

Thomas Zaslavsky, Mar 14 2010

Keywords

• nonn
• easy

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

• Thomas Zaslavsky, Table of n, a(n) for n = 1..10000.
• 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.
• Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1).

Crossrefs

Cf. A108576, A108577, A174257.

Programs

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 *)

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