A335606 The number of fixed n-ominoes with a convex hull of width 3.

1, 8, 31, 95, 269, 721, 1866, 4728, 11804, 29162, 71502, 174342, 423341, 1024786, 2474934, 5966625, 14365256, 34550674, 83035396, 199440433, 478814076, 1149133511, 2757142136, 6613933242, 15863281135, 38042981575, 91225540813, 218739876078, 524464594304, 1257437814143, 3014693395137

Offset

3,2

Comments

Obtained from Zeilberger's tables by subtracting the numbers of width <= 3 and of width <= 2.

Links

Table of n, a(n) for n=3..33.

D. Zeilberger, Series expansion for 2D lattice-animals of globally bounded width

Index entries for sequences related to polyominoes

Index entries for linear recurrences with constant coefficients, signature (5,-6,-4,8,1,2,-8,0,-1,9,-2,-1,-3,1).

Formula

a(n) = A308359(n,3).

G.f.: -x^3*(1+x) *(x^10 -x^9 -3*x^8 +2*x^7 -x^6 -2*x^5 +7*x^4 -3*x^3 -5*x^2 +2*x +1) / ( (x-1) *(x^3 +x^2 +x -1) *(x^10 -3*x^9 -x^8 +2*x^6 +x^4 -4*x^3 +3*x -1) ).

a(n)= 5*a(n-1) -6*a(n-2) -4*a(n-3) +8*a(n-4) +a(n-5) +2*a(n-6) -8*a(n-7) -a(n-9) +9*a(n-10) -2*a(n-11) -a(n-12) -3*a(n-13) +a(n-14).

Example

a(3)=1 counts 1 3-omino of shape 1x3.
a(4)=8 counts 8 4-ominoes of shape 2x3.
a(5)=31 counts 6 5-ominoes of shape 2x3 and 25 5-ominoes of shape 3x3.
a(6)=95 counts 1 6-omino of shape 2x3, 44 6-ominoes of shape 3x3 and 50 6-ominoes of shape 4x3.

Mathematica

LinearRecurrence[{5, -6, -4, 8, 1, 2, -8, 0, -1, 9, -2, -1, -3, 1}, {1, 8, 31, 95, 269, 721, 1866, 4728, 11804, 29162, 71502, 174342, 423341, 1024786, 2474934}, 31] (* Georg Fischer, Jan 16 2021 *)

Crossrefs

Cf. A308359, A027053 (width 2).

Sequence in context: A319906 A212064 A213764 * A320416 A289613 A055845

Adjacent sequences: A335603 A335604 A335605 * A335607 A335608 A335609

Keyword

nonn,easy

Author

R. J. Mathar, Jun 15 2020

Status

approved