

A108578


Number of 3 X 3 magic squares with magic sum 3n.


7



0, 0, 0, 0, 8, 24, 32, 56, 80, 104, 136, 176, 208, 256, 304, 352, 408, 472, 528, 600, 672, 744, 824, 912, 992, 1088, 1184, 1280, 1384, 1496, 1600, 1720, 1840, 1960, 2088, 2224, 2352, 2496, 2640, 2784, 2936, 3096, 3248, 3416, 3584, 3752, 3928, 4112, 4288
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OFFSET

1,5


COMMENTS

Contribution from Thomas Zaslavsky, Mar 12 2010: (Start)
A magic square has distinct positive integers in its cells, whose sum is the same (the "magic sum") along any row, column, or main diagonal.
a(n) is given by a quasipolynomial of period 6. (End)


LINKS

T. Zaslavsky, Table of n, a(n) for n = 1..10000.
M. Beck and T. Zaslavsky, An enumerative geometry for magic and magilatin labellings, Ann. Combinatorics, 10 (2006), no. 4, 395413. MR 2007m:05010. Zbl 1116.05071.  Thomas Zaslavsky, Jan 29 2010
Matthias Beck and Thomas Zaslavsky, Six Little Squares and How their Numbers Grow, Journal of Integer Sequences, 13 (2010), Article 10.6.2.Index entries for linear recurrences with constant coefficients, signature (1,1,0,1, 1,1).


FORMULA

G.f.: [8*x^5*(1+2*x)] / [(1x)*(1x^2)*(1x^3)].
a(n) = a(n1) + a(n2)  a(n4)  a(n5) + a(n6).  Vincenzo Librandi, Sep 01 2018


EXAMPLE

a(5) = 8 because there are 8 3 X 3 magic squares with entries 1,...,9 and magic sum 15.


MATHEMATICA

LinearRecurrence[{1, 1, 0, 1, 1, 1}, {0, 0, 0, 0, 8, 24}, 50] (* JeanFrançois Alcover, Sep 01 2018 *)
CoefficientList[Series[8 x^4 (1 + 2 x) / ((1  x) (1  x^2) (1  x^3)), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 01 2018 *)


PROG

(PARI) a(n)=(1/9)*(2*n^232*n+[144, 78, 120, 126, 96, 102][(n%18)/3+1])
(PARI) x='x+O('x^99); concat(vector(4), Vec(8*x^5*(1+2*x)/((1x)*(1x^2)*(1x^3)))) \\ Altug Alkan, Sep 01 2018
(MAGMA) I:=[0, 0, 0, 0, 8, 24]; [n le 6 select I[n] else Self(n1)+Self(n2)Self(n4)Self(n5)+Self(n6): n in [1..60]]; // Vincenzo Librandi, Sep 01 2018


CROSSREFS

Equals 8 times the second differences of A055328.
Cf. A108576, A108577, A108579.
Sequence in context: A283078 A319528 A140403 * A305241 A044450 A134223
Adjacent sequences: A108575 A108576 A108577 * A108579 A108580 A108581


KEYWORD

nonn


AUTHOR

Thomas Zaslavsky and Ralf Stephan, Jun 11 2005


EXTENSIONS

Edited by N. J. A. Sloane, Feb 05 2010
Corrected g.f. to account for previous change in parameter n from magic sum s to s/3; by Thomas Zaslavsky, Mar 12 2010


STATUS

approved



