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

1, 1, 1, 3, 5, 11, 9, 26, 22, 53, 36, 93, 64, 151, 94, 228, 143, 329, 195, 455, 271, 611, 351, 798, 460, 1021, 574, 1281, 722, 1583, 876, 1928, 1069, 2321, 1269, 2763, 1513, 3259, 1765, 3810, 2066, 4421, 2376, 5093, 2740, 5831, 3114, 6636, 3547, 7513, 3991

Offset

1,4

Links

Colin Barker, Table of n, a(n) for n = 1..1000

K. Bezdek, P. Brass and H. Harborth, Maximum convex hulls of connected systems of segments and of polyominoes, Beiträge Algebra Geom., Vol. 35(1) (1994), pp. 37-43.

S. Kurz, Polyominoes with maximum convex hull, Diploma thesis, Bayreuth (2004).

Index entries for linear recurrences with constant coefficients, signature (0,2,0,1,0,-4,0,1,0,2,0,-1).

Formula

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

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

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

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

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).

From Luce ETIENNE, Aug 14 2019: (Start)

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

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

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)

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

Maple

A122133 := proc(n)
    if modp(n, 4)= 0 then
        (n^3-2*n^2+4*n)/16 ;
    elif modp(n, 4)= 1 then
        (n^3-2*n^2+13*n+20)/32 ;
    elif modp(n, 4)= 2 then
        (n^3-2*n^2+4*n+8)/16 ;
    else
        (n^3-2*n^2+5*n+8)/32 ;
    fi;
end proc: # R. J. Mathar, May 19 2019

Prog

(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

Crossrefs

Sequence in context: A105603 A264987 A170835 * A273377 A065019 A071328

Adjacent sequences: A122130 A122131 A122132 * A122134 A122135 A122136

Keyword

nonn,easy

Author

Sascha Kurz, Aug 21 2006

Status

approved