A053091 - OEIS

%I #23 Oct 13 2013 03:28:37

%S 1,3,5,6,9,11,10,15,18,14,21,23,18,30,29,21,33,35,31,39,41,30,42,54,

%T 35,51,53,38,66,54,42,63,65,60,69,70,43,75,90,54,81,83,63,93,89,62,90,

%U 95,84,99,90,77,105,126,74,111,113,60,138,119,91,126,125,108

%N F^3-convex polyominoes on the honeycomb lattice by number of cells.

%C 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

%D 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.

%D 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.

%H Alain Denise, Christoph Duerr and Fouad Ibn-Majdoub-Hassani <a href="http://www.fpsac.org/FPSAC97/ARTICLES/Denise.ps.gz">Enumeration et generation aleatoire de polyominos convexes en reseau hexagonal (French)</a>

%F 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

%F G.f.: sum_{n >= 1} sum{d|n} b_d^2 * x^d * (1 + sign(n-d)), where b_0 = 0 and

%F b_i = x^binomial(i, 2) * sum_{k=1}^{i} x^(-binomial(i, 2)) for i >= 1 [corrected by _Michael Somos_, Jun 21 2012]

%e 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 + ...

%e +---+

%e | o | a(1) = 1

%e +---------------+

%e | o o | o  |  o | a(2) = 3

%e |     |  o | o  |

%e +-------------------------------+

%e |  o  | o o |       | o   |   o |

%e | o o |  o  | o o o |  o  |  o  | a(3) = 5

%e |     |     |       |   o | o   |

%e +-------------------------------------------+

%e |         | o    |    o |  o  |      |      |

%e | o o o o |  o   |   o  | o o |  o o | o o  | a(4) = 6

%e |         |   o  |  o   |  o  | o o  |  o o |

%e |         |    o | o    |     |      |      |

%e +-------------------------------------------+

%e - _Michael Somos_, Jun 21 2012

%o (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 */

%o (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 */

%K nonn

%O 1,2

%A _Fouad IBN MAJDOUB HASSANI_, Feb 28 2000