A053090 Number of F^3-convex polyominoes on honeycomb lattice with given semiperimeter.

1, 0, 3, 2, 6, 6, 12, 12, 21, 22, 33, 36, 50, 54, 72, 78, 99, 108, 133, 144, 174, 188, 222, 240, 279, 300, 345, 370, 420, 450, 506, 540, 603, 642, 711, 756, 832, 882, 966, 1022, 1113, 1176, 1275, 1344, 1452, 1528, 1644, 1728, 1853, 1944, 2079, 2178, 2322, 2430

Offset: 3

Author

Fouad IBN MAJDOUB HASSANI, Feb 28 2000

Keywords

• nonn
• easy

Comments

Sequence is also given by the Poincaré series [or Poincare series] of an ordinal Hodge algebra, or algebra with straightening law, that the three-strand braid group acts on. - Stephen P. Humphries, Feb 06 2009

From Michael Somos, Jun 21 2012: (Start)

Euler transform of length-6 sequence [ 0, 3, 2, 0, 0, -1].

Expansion of F^3(x, 1, 1, 1) in powers of x where F^3(x, y, q, t) is the generating function defined in the FPSAC97 article.

The polyominoes are counted up to translations but not rotations and reflections. Thus, the unique domino with two cells is counted three times for its three orientations.

The semiperimeter of each hexagonal cell is 3 but each common side shared by two cells decreases the semiperimeter by one. (End)

From Jianing Song, Apr 27 2022: (Start)

Let w = exp(2*Pi*i/6) be a primitive 6th root of 1. Then a(n+3) is number of nonnegative integer solutions to Sum_{j=0..5} x_j = n, Sum_{j=0..5} (x_j)*w^j = 0.

Proof: Sum_{j=0..5} (x_j)*w^j = x_0 + (x_1)*w + (x_2)*(w-1) - x_3 - (x_4)*w - (x_5)*(w-1), so Sum_{j=0..5} (x_j)*w^j = 0 if and only if x_0 + x_5 = x_2 + x_3 and x_1 + x_2 = x_4 + x_5, or equivalently, x_0 - x_3 = x_2 - x_5 = x_4 - x_1.

Case (a): x_0 - x_3 = x_2 - x_5 = x_4 - x_1 >= 0, then we can write x_0 = x+t, x_1 = z, x_2 = y+t, x_3 = x, x_4 = z+t, x_5 = y. The number of solutions in this case is equal to the number of solutions to 2*s + 3*t = n, x+y+z = s.

Case (b): x_0 - x_3 = x_2 - x_5 = x_4 - x_1 <= 0, then we can write x_0 = x, x_1 = z+t, x_2 = y, x_3 = x+t, x_4 = z, x_5 = y+t. The number of solutions in this case is also equal to the number of solutions to 2*s + 3*t = n, x+y+z = s.

The common part of case (a) and case (b) is the case where x_0 - x_3 = x_2 - x_5 = x_4 - x_1 = 0, in which case the number of solutions is equal to the number of solutions to x+y+z = n/2 for even n and 0 for odd n.

In conclusion, the total number of solutions is 2*Sum_{s=0..n/2, 3|(n-2*s)} (s+1)*(s+2)/2 - [n even]*(n/2+1)*(n/2+2)/2, where [] is an Iverson bracket. This can be shown to be equal to a(n+3). (End)

Examples

			x^3 + 3*x^5 + 2*x^6 + 6*x^7 + 6*x^8 + 12*x^9 + 12*x^10 + 21*x^11 + ...
+---+
| o | a(3) = 1
+---------------+
| o o | o  |  o | a(5) = 3
|     |  o | o  |
+---------------+
|  o  | o o | a(6) = 2
| o o |  o  |
+---------------------------------------+
|       | o   |   o |  o  |      | o o  |
| o o o |  o  |  o  | o o |  o o |  o o | a(7) = 6
|       |   o | o   |  o  | o o  |      |
+---------------------------------------+
- _Michael Somos_, Jun 21 2012
		

References

• Fouad Ibn-Majdoub-Hassani. Combinatoire de polyominos et des tableaux décalés oscillants. Thèse de Doctorat. Laboratoire de Recherche en Informatique, Université Paris-Sud XI, France.
• Alain Denise, Christoph Durr and Fouad Ibn-Majdoub-Hassani. Enumération et génération aléatoire de polyominos convexes en réseau hexagonal (French) [enumeration and random generation of convex polyominoes in the honeycomb lattice]. In Proceedings of 9th Conference on Formal Power Series and Algebraic Combinatorics (FPSAC97), pages 222-234, 1997.

Links

• Jianing Song, Table of n, a(n) for n = 3..10003
• Alain Denise, Christoph Duerr and Fouad Ibn-Majdoub-Hassani, Enumération et génération aléatoire de polyominos convexes en réseau hexagonal (French)
• Stephen P. Humphries, Action of some braid groups on Hodge algebras Comm. Algebra 26 (1998), no. 4, pages 1233-1242. See Proposition 3.4 on page 1241. [From _Stephen P. Humphries_, Feb 06 2009]
• Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-2,-1,2,1,-1)

Programs

Maple

A053090 := proc(n)
    n*(2*n^2+3-27*(-1)^n)-32*A049347(n-1) ;
    %/144 ;
end proc:
seq(A053090(n),n=3..30) ; # R. J. Mathar, Mar 12 2025

PARI

{a(n) = round( n * (2*n^2 + 3) / 144 - (-1)^n * 3*n / 16)} /* Michael Somos, Jun 21 2012 */

PARI

{a(n) = sign(n) * polcoeff( x^3 * (1 + x^3) / ((1 - x^2)^3 * (1 - x^3)) + x * O(x^abs(n)), abs(n))} /* Michael Somos, Jun 21 2012 */

Formula

G.f.: x^3*(1 + x^3)/((1 - x^2)^3*(1 - x^3)).

a(-n) = -a(n). a(n) = round( n*(2*n^2 + 3)/144 - (-1)^n*3*n/16 ). - Michael Somos, Jun 21 2012