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A053090
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Number of F^3-convex polyominoes on honeycomb lattice with given semiperimeter.
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4
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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
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OFFSET
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3,3
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COMMENTS
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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)
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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 (FPSAC97), pages 222-234, 1997.
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LINKS
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Table of n, a(n) for n=3..56.
Alain Denise, Christoph Duerr and Fouad Ibn-Majdoub-Hassani Enumeration et generation aleatoire de polyominos convexes en reseau 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)
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FORMULA
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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
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EXAMPLE
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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
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PROG
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(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 */
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CROSSREFS
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Sequence in context: A062200 A114208 A014686 * A264400 A225367 A283479
Adjacent sequences: A053087 A053088 A053089 * A053091 A053092 A053093
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KEYWORD
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nonn,easy
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AUTHOR
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Fouad IBN MAJDOUB HASSANI, Feb 28 2000
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STATUS
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approved
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