|
%I #21 Nov 12 2018 03:04:47
%S 0,0,0,0,0,0,0,0,0,8,16,40,64,96,128,184,240,320,400,504,608,744,880,
%T 1056,1232,1440,1648,1904,2160,2464,2768,3120,3472,3880,4288,4760,
%U 5232,5760,6288,6888,7488,8160,8832,9576,10320,11144,11968,12880,13792,14784,15776
%N Number of 3 X 3 magic squares (with distinct positive entries) having all entries < n.
%C From _Thomas Zaslavsky_, Mar 12 2010: (Start)
%C 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.
%C a(n) is given by a quasipolynomial of period 12. (End)
%H T. Zaslavsky, <a href="/A108576/b108576.txt">Table of n, a(n) for n = 1..10000</a>.
%H M. Beck and T. Zaslavsky, <a href="https://arxiv.org/abs/math/0506315">An enumerative geometry for magic and magilatin labellings</a>, arXiv:math/0506315 [math.CO], 2005; Ann. Combinatorics, 10 (2006), no. 4, 395-413. MR 2007m:05010. Zbl 1116.05071. - _Thomas Zaslavsky_, Jan 29 2010
%H M. Beck and T. Zaslavsky, <a href="http://www.math.binghamton.edu/zaslav/Tpapers/index.html#SLS">Six little squares and how their numbers grow</a>, submitted. - _Thomas Zaslavsky_, Jan 29 2010
%H Matthias Beck and Thomas Zaslavsky, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Zaslavsky/sls.html">Six Little Squares and How their Numbers Grow</a>, Journal of Integer Sequences, 13 (2010), Article 10.6.2.
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,1,-2,2,-2,1,0,-1,2,-1).
%F G.f.: (8*x^10*(2*x^2+1)) / ((1-x^6)*(1-x^4)*(1-x)^2).
%F a(n) is given by a quasipolynomial of period 12.
%e a(10) = 8 because there are 8 3 X 3 magic squares with distinct entries < 10 (they are the standard magic squares).
%t LinearRecurrence[{2, -1, 0, 1, -2, 2, -2, 1, 0, -1, 2, -1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 16, 40}, 60] (* _Jean-François Alcover_, Nov 12 2018 *)
%t CoefficientList[Series[(8 x^10 (2 x^2 + 1)) / ((1 - x^6) (1 - x^4) (1 - x)^2), {x, 0, 60}], x] (* _Vincenzo Librandi_, Nov 12 2018 *)
%o (PARI) a(n)=1/6*(n^3-16*n^2+(76-3*(n%2))*n -[96,58,96,102,112,90,96,70,96,90,112,102][(n%12)+1])
%Y Cf. A108577, A108578, A108579.
%K nonn,easy
%O 1,10
%A _Thomas Zaslavsky_ and _Ralf Stephan_, Jun 11 2005
%E Edited by _N. J. A. Sloane_, Feb 05 2010
|
