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A053091
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F^3-convex polyominoes on the honeycomb lattice by number of cells.
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1
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1, 3, 5, 6, 9, 11, 10, 15, 18, 14, 21, 23, 18, 30, 29, 21, 33, 35, 31, 39, 41, 30, 42, 54, 35, 51, 53, 38, 66, 54, 42, 63, 65, 60, 69, 70, 43, 75, 90, 54, 81, 83, 63, 93, 89, 62, 90, 95, 84, 99, 90, 77, 105, 126, 74, 111, 113, 60, 138, 119, 91, 126, 125, 108
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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FORMULA
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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]
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EXAMPLE
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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 | | | |
+-------------------------------------------+
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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