OFFSET
1,2
COMMENTS
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. - Michael Somos, Jun 21 2012
REFERENCES
Fouad Ibn-Majdoub-Hassani. Combinatoire de polyominos et des tableaux decales oscillants. These de Doctorat. Laboratoire de Recherche en Informatique, Universite Paris-Sud XI, France.
Alain Denise, Christoph Durr and Fouad Ibn-Majdoub-Hassani. Enumeration et generation aleatoire de polyominos convexes en reseau 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, pages 222-234, 1997.
LINKS
Alain Denise, Christoph Duerr and Fouad Ibn-Majdoub-Hassani Enumeration et generation aleatoire de polyominos convexes en reseau hexagonal (French)
FORMULA
Expansion of F^3(1, 1, q, 1) in powers of q where F^3(x, y, q, t) is the generating function defined in the FPSAC97 article. - Michael Somos, Jun 20 2012
G.f.: sum_{n >= 1} sum{d|n} b_d^2 * x^d * (1 + sign(n-d)), where b_0 = 0 and
b_i = x^binomial(i, 2) * sum_{k=1}^{i} x^(-binomial(i, 2)) for i >= 1 [corrected by Michael Somos, Jun 21 2012]
EXAMPLE
x + 3*x^2 + 5*x^3 + 6*x^4 + 9*x^5 + 11*x^6 + 10*x^7 + 15*x^8 + 18*x^9 + ...
+---+
| o | a(1) = 1
+---------------+
| o o | o | o | a(2) = 3
| | o | o |
+-------------------------------+
| o | o o | | o | o |
| o o | o | o o o | o | o | a(3) = 5
| | | | o | o |
+-------------------------------------------+
| | o | o | o | | |
| o o o o | o | o | o o | o o | o o | a(4) = 6
| | o | o | o | o o | o o |
| | o | o | | | |
+-------------------------------------------+
- Michael Somos, Jun 21 2012
PROG
(PARI) {a(n) = local(m = 4*n); if( n<1, 0, (-1)^n / 2 * polcoeff( sum( k=1, m, k * kronecker( 2, k) * if( k%4 == 3, x^k, x^(3*k)) / (1 + x^(4*k)), O(x^m)), m - 1))} /* Michael Somos, Jun 20 2012 */
(PARI) {a(n) = if( n<1, 0, polcoeff( sum( i=1, n, x^i * (1 + x^i) / (1 - x^i) * ( sum( k=1, i, x^((i - k) * (i + k - 1)/2), x * O(x^(n - i))))^2 ), n))} /* Michael Somos, Jun 21 2012 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Fouad IBN MAJDOUB HASSANI, Feb 28 2000
STATUS
approved