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A241526 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.

%I A241526 #29 Jun 13 2015 00:55:01
%S A241526 2,7,16,31,53,83,123,174,237,314,406,514,640,785,950,1137,1347,1581,
%T A241526 1841,2128,2443,2788,3164,3572,4014,4491,5004,5555,6145,6775,7447,
%U A241526 8162,8921,9726,10578,11478,12428,13429,14482,15589,16751,17969,19245,20580,21975
%N A241526 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 A241526 Christopher Hunt Gribble, <a href="/A241526/b241526.txt">Table of n, a(n) for n = 1..10000</a>
%H A241526 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2,-3,3,-1).
%F A241526 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 A241526 a(n) = (4*n^3+15*n^2+17*n-6*floor(n/3))/18.
%F A241526 G.f.: x*(x^2+x+2) / ((x-1)^4*(x^2+x+1)). - _Colin Barker_, Apr 26 2014
%e A241526 The bi-symmetric triangle of 1 X 1 squares of height 5 is:
%e A241526                    ___
%e A241526                  _|_|_|_
%e A241526                _|_|_|_|_|_
%e A241526              _|_|_|_|_|_|_|_
%e A241526            _|_|_|_|_|_|_|_|_|_
%e A241526           |_|_|_|_|_|_|_|_|_|_|
%e A241526 .
%e A241526 No. of positions in which a 1 X 1 square can be placed = 2 + 4 + 6 + 8 + 10 = 30.
%e A241526 No. of positions in which a 2 X 2 square can be placed = 1 + 3 + 5 + 7 = 16.
%e A241526 No. of positions in which a 3 X 3 square can be placed = 2 + 4 = 6.
%e A241526 No. of positions in which a 4 X 4 square can be placed = 1.
%e A241526 Thus, a(5) = 30 + 16 + 6 + 1 = 53.
%p A241526 a := proc (n::integer)::integer;
%p A241526        (2/9)*n^3+(5/6)*n^2+(17/18)*n-(1/3)*floor((1/3)*n)
%p A241526      end proc:
%p A241526 seq(a(n), n = 1..60);
%o A241526 (PARI) Vec(x*(x^2+x+2)/((x-1)^4*(x^2+x+1)) + O(x^100)) \\ _Colin Barker_, Apr 26 2014
%Y A241526 Cf. A092498.
%K A241526 nonn,easy
%O A241526 1,1
%A A241526 _Christopher Hunt Gribble_ and _Luce ETIENNE_, Apr 24 2014