A093119 Number of convex polyominoes with a 3 X n+1 minimal bounding rectangle.

13, 68, 222, 555, 1171, 2198, 3788, 6117, 9385, 13816, 19658, 27183, 36687, 48490, 62936, 80393, 101253, 125932, 154870, 188531, 227403, 271998, 322852, 380525, 445601, 518688, 600418, 691447, 792455, 904146, 1027248, 1162513

Offset

1,1

Links

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

V. J. W. Guo and J. Zeng, The number of convex polyominoes and the generating function of Jacobi polynomials, arXiv:math/0403262 [math.CO], 2004.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = ((3*n+2)*C(2n+4, 4) - 4*n*C(n+2, n)^2)/(n+2), n>0.

a(n) = (6*n^4 + 20*n^3 + 27*n^2 + 19*n + 6)/6.

From Colin Barker, Feb 24 2019: (Start)

G.f.: x*(13 + 3*x + 12*x^2 - 5*x^3 + x^4) / (1 - x)^5.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.

(End)

E.g.f.: -1 + (6 + 72*x + 129*x^2 + 56*x^3 + 6*x^4)*exp(x)/6. - G. C. Greubel, Jun 26 2019

Mathematica

a[n_] := n^4 + 10*n^3/3 + 9*n^2/2 + 19*n/6 + 1; Array[a, 40] (* Jean-François Alcover, Feb 24 2019 *)

Prog

(PARI) Vec(x*(13 + 3*x + 12*x^2 - 5*x^3 + x^4) / (1 - x)^5 + O(x^40)) \\ Colin Barker, Feb 24 2019
(Magma) [(6*n^4 + 20*n^3 + 27*n^2 + 19*n + 6)/6: n in [1..40]]; // G. C. Greubel, Jun 26 2019
(Sage) [(6*n^4 + 20*n^3 + 27*n^2 + 19*n + 6)/6 for n in (1..40)] # G. C. Greubel, Jun 26 2019
(GAP) List([1..40], n-> (6*n^4 + 20*n^3 + 27*n^2 + 19*n + 6)/6) # G. C. Greubel, Jun 26 2019

Crossrefs

Row 2 of triangle A093118.

Sequence in context: A213355 A229999 A258618 * A362102 A239538 A156794

Adjacent sequences: A093116 A093117 A093118 * A093120 A093121 A093122

Keyword

nonn,easy

Author

Ralf Stephan, Mar 21 2004

Status

approved