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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 (list; graph; refs; listen; history; text; internal format)
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) = 8*A174257.

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

T. Zaslavsky, Table of n, a(n) for n = 1..10000.

M. Beck, T. 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

G.f.: 8 * x^8 * (2*x^2+1) / ((x^4-1) * (x^6-1)).

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

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

Cf. A108576, A108577, A174257.

Sequence in context: A073925 A053321 A299214 * A037239 A205869 A217178

Adjacent sequences:  A174253 A174254 A174255 * A174257 A174258 A174259

KEYWORD

nonn,easy

AUTHOR

Thomas Zaslavsky, Mar 14 2010

STATUS

approved

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Last modified October 15 15:14 EDT 2019. Contains 328030 sequences. (Running on oeis4.)