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A122133 Number of different polyominoes with maximum area of the convex hull.

%I A122133 #37 Aug 23 2019 04:02:41
%S A122133 1,1,1,3,5,11,9,26,22,53,36,93,64,151,94,228,143,329,195,455,271,611,
%T A122133 351,798,460,1021,574,1281,722,1583,876,1928,1069,2321,1269,2763,1513,
%U A122133 3259,1765,3810,2066,4421,2376,5093,2740,5831,3114,6636,3547,7513,3991
%N A122133 Number of different polyominoes with maximum area of the convex hull.
%H A122133 Colin Barker, <a href="/A122133/b122133.txt">Table of n, a(n) for n = 1..1000</a>
%H A122133 K. Bezdek, P. Brass and H. Harborth, <a href="http://eudml.org/doc/232207">Maximum convex hulls of connected systems of segments and of polyominoes</a>, Beiträge Algebra Geom., Vol. 35(1) (1994), pp. 37-43.
%H A122133 S. Kurz, <a href="http://www.wm-archive.uni-bayreuth.de/fileadmin/Sascha/Publikationen2/thesis.pdf">Polyominoes with maximum convex hull</a>, Diploma thesis, Bayreuth (2004).
%H A122133 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,1,0,-4,0,1,0,2,0,-1).
%F A122133 a(n) = (n^3 - 2*n^2 + 4*n)/16 if n mod 4 = 0;
%F A122133 a(n) = (n^3 - 2*n^2 + 13*n + 20)/32 if n mod 4 = 1;
%F A122133 a(n) = (n^3 - 2*n^2 + 4*n + 8)/16 if n mod 4 = 2;
%F A122133 a(n) = (n^3 - 2*n^2 + 5*n + 8)/32 if n mod 4 = 3.
%F A122133 G.f.: (1 + x - x^2 - x^3 + 2*x^5 + 8*x^6 + 2*x^7 + 4*x^8 + 2*x^9 - x^10 + x^12)/((1-x^2)^2*(1-x^4)^2).
%F A122133 From _Luce ETIENNE_, Aug 14 2019: (Start)
%F A122133 a(n) = 4*a(n-4) - 6*a(n-8) + 4*a(n-12) - a(n-16).
%F A122133 a(n) = 2*a(n-2) + a(n-4) - 4*a(n-6) + a(n-8) + 2*a(n-10) - a(n-12).
%F A122133 a(n) = (3*n^3 - 6*n^2 + 17*n + 22 + (n^3 - 2*n^2 - n - 6)*(-1)^n - 4*(4*cos(n*Pi/2) - (2*n+3)*sin(n*Pi/2)))/64. (End)
%F A122133 E.g.f.: (1/64)*(-exp(-x)*(6 - 2*x - x^2 + x^3) + exp(x)*(22 + 14*x + 3*x^2 + 3*x^3) - 4*(4*cos(x) - 2*x*cos(x) - 3*sin(x))). - _Stefano Spezia_, Aug 14 2019
%p A122133 A122133 := proc(n)
%p A122133     if modp(n,4)= 0 then
%p A122133         (n^3-2*n^2+4*n)/16 ;
%p A122133     elif modp(n,4)= 1 then
%p A122133         (n^3-2*n^2+13*n+20)/32 ;
%p A122133     elif modp(n,4)= 2 then
%p A122133         (n^3-2*n^2+4*n+8)/16 ;
%p A122133     else
%p A122133         (n^3-2*n^2+5*n+8)/32 ;
%p A122133     fi;
%p A122133 end proc: # _R. J. Mathar_, May 19 2019
%o A122133 (PARI) Vec(x*(1+x-x^2+x^3+2*x^4+4*x^5+2*x^6+5*x^7+2*x^8+x^9)/((1-x)^4*(1+x)^4*(1+x^2)^2) + O(x^80)) \\ _Colin Barker_, Oct 14 2016
%K A122133 nonn,easy
%O A122133 1,4
%A A122133 _Sascha Kurz_, Aug 21 2006