A053091 F^3-convex polyominoes on the honeycomb lattice by number of cells.
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
Offset: 1
Keywords
Examples
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
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)
Programs
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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 */
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]
Comments