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Number of different positions in which a square with side length k, 1 <= k <= n - floor(n/3), can be placed within a bi-symmetric triangle of 1 X 1 squares of height n.
3

%I #29 Jun 13 2015 00:55:01

%S 2,7,16,31,53,83,123,174,237,314,406,514,640,785,950,1137,1347,1581,

%T 1841,2128,2443,2788,3164,3572,4014,4491,5004,5555,6145,6775,7447,

%U 8162,8921,9726,10578,11478,12428,13429,14482,15589,16751,17969,19245,20580,21975

%N Number of different positions in which a square with side length k, 1 <= k <= n - floor(n/3), can be placed within a bi-symmetric triangle of 1 X 1 squares of height n.

%H Christopher Hunt Gribble, <a href="/A241526/b241526.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2,-3,3,-1).

%F a(n) = sum_{j=0..n-1-floor(n/3)} ((4*n-6*j+1-(-1)^j)/4)*((4*n-6*j+3+(-1)^j)/4).

%F a(n) = (4*n^3+15*n^2+17*n-6*floor(n/3))/18.

%F G.f.: x*(x^2+x+2) / ((x-1)^4*(x^2+x+1)). - _Colin Barker_, Apr 26 2014

%e The bi-symmetric triangle of 1 X 1 squares of height 5 is:

%e ___

%e _|_|_|_

%e _|_|_|_|_|_

%e _|_|_|_|_|_|_|_

%e _|_|_|_|_|_|_|_|_|_

%e |_|_|_|_|_|_|_|_|_|_|

%e .

%e No. of positions in which a 1 X 1 square can be placed = 2 + 4 + 6 + 8 + 10 = 30.

%e No. of positions in which a 2 X 2 square can be placed = 1 + 3 + 5 + 7 = 16.

%e No. of positions in which a 3 X 3 square can be placed = 2 + 4 = 6.

%e No. of positions in which a 4 X 4 square can be placed = 1.

%e Thus, a(5) = 30 + 16 + 6 + 1 = 53.

%p a := proc (n::integer)::integer;

%p (2/9)*n^3+(5/6)*n^2+(17/18)*n-(1/3)*floor((1/3)*n)

%p end proc:

%p seq(a(n), n = 1..60);

%o (PARI) Vec(x*(x^2+x+2)/((x-1)^4*(x^2+x+1)) + O(x^100)) \\ _Colin Barker_, Apr 26 2014

%Y Cf. A092498.

%K nonn,easy

%O 1,1

%A _Christopher Hunt Gribble_ and _Luce ETIENNE_, Apr 24 2014