A122133 - OEIS

%I #37 Aug 23 2019 04:02:41

%S 1,1,1,3,5,11,9,26,22,53,36,93,64,151,94,228,143,329,195,455,271,611,

%T 351,798,460,1021,574,1281,722,1583,876,1928,1069,2321,1269,2763,1513,

%U 3259,1765,3810,2066,4421,2376,5093,2740,5831,3114,6636,3547,7513,3991

%N Number of different polyominoes with maximum area of the convex hull.

%H Colin Barker, <a href="/A122133/b122133.txt">Table of n, a(n) for n = 1..1000</a>

%H 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 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 <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 a(n) = (n^3 - 2*n^2 + 4*n)/16 if n mod 4 = 0;

%F a(n) = (n^3 - 2*n^2 + 13*n + 20)/32 if n mod 4 = 1;

%F a(n) = (n^3 - 2*n^2 + 4*n + 8)/16 if n mod 4 = 2;

%F a(n) = (n^3 - 2*n^2 + 5*n + 8)/32 if n mod 4 = 3.

%F 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 From _Luce ETIENNE_, Aug 14 2019: (Start)

%F a(n) = 4*a(n-4) - 6*a(n-8) + 4*a(n-12) - a(n-16).

%F a(n) = 2*a(n-2) + a(n-4) - 4*a(n-6) + a(n-8) + 2*a(n-10) - a(n-12).

%F 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 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 := proc(n)

%p     if modp(n,4)= 0 then

%p         (n^3-2*n^2+4*n)/16 ;

%p     elif modp(n,4)= 1 then

%p         (n^3-2*n^2+13*n+20)/32 ;

%p     elif modp(n,4)= 2 then

%p         (n^3-2*n^2+4*n+8)/16 ;

%p     else

%p         (n^3-2*n^2+5*n+8)/32 ;

%p     fi;

%p end proc: # _R. J. Mathar_, May 19 2019

%o (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 nonn,easy

%O 1,4

%A _Sascha Kurz_, Aug 21 2006