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

Author

Ralf Stephan, Mar 21 2004

Keywords

• nonn
• easy

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).

Crossrefs

Row 2 of triangle A093118.

Programs

GAP

List([1..40], n-> (6*n^4 + 20*n^3 + 27*n^2 + 19*n + 6)/6); # G. C. Greubel, Jun 26 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

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 *)

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

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

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