

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
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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^4 * (2*x+1) / ((x^21) * (x^31)). [amended by Georg Fischer, Apr 17 2020]
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^41)(x^61)), {x, 0, 120}], x], {1, 1, 2}] (* Harvey P. Dale, Aug 07 2017 *)


CROSSREFS

Cf. A108576, A108577, A174257.
Sequence in context: A053321 A335772 A299214 * A037239 A205869 A217178
Adjacent sequences: A174253 A174254 A174255 * A174257 A174258 A174259


KEYWORD

nonn,easy


AUTHOR

Thomas Zaslavsky, Mar 14 2010


STATUS

approved



