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A122133 Number of different polyominoes with maximum area of the convex hull. 2
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 (list; graph; refs; listen; history; text; internal format)
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, Beitr├Ąge 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

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Last modified March 29 02:19 EDT 2020. Contains 333104 sequences. (Running on oeis4.)