%I #19 May 02 2023 08:36:16

%S 1,0,0,0,0,0,0,0,0,0,2,0,0,0,2,0,2,2,8,0,18,6,16,6,48,22,74,48,182,74,

%T 306,204,544,342,1114,826,2038,1546,4144,3126,7452,6470,14538,12542,

%U 27824,25994,53398,50244,103288,101306,195756,200120,380310,395802

%N Number of tilings of a 5 X n rectangle using n pentominoes of shapes T, N, X.

%H Alois P. Heinz, <a href="/A361250/b361250.txt">Table of n, a(n) for n = 0..5000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Pentomino">Pentomino</a>

%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1,1,0,1,0,3,-1,6,-2,0,-6,-1).

%e a(10) = 2:

%e .___________________.

%e |___. |_. ._| .___| |

%e |_. |___| |___|___ |

%e | |_____|_| |___. |_|

%e | .___| ._| |_. |___|

%e |_|_____|_____|_____| ... and its mirror.

%e .

%e a(14) = 2:

%e .___________________________.

%e |___. |_. ._| |_. ._| .___| |

%e |_. |___| |_. ._| |___|___ |

%e | |_____|_| |_| |_| |___. |_|

%e | .___| ._| |_. ._| |_. |___|

%e |_|_____|_____|_|_____|_____| ... and its mirror.

%e .

%e a(16) = 2:

%e ._______________________________.

%e |___. |_. ._| .___|_. ._| .___| |

%e |_. |___| |___| .___| |___|___ |

%e | |_____|_| |___| ._|_| |___. |_|

%e | .___| ._| |_____| ._| |_. |___|

%e |_|_____|_____|_____|_____|_____| ... and its mirror.

%e .

%e a(17) = 2:

%e ._________________________________.

%e |___. |_. ._| |___. |_. ._| .___| |

%e |_. |___| |_. ._| |___| |___|___ |

%e | |_____|_| |_|_. ._| |_| |___. |_|

%e | .___| ._| |_. |_|_. ._| |_. |___|

%e |_|_____|_____|_____|_|_____|_____| ... and its mirror.

%e .

%p gf:= (x^14+4*x^13+2*x^11-4*x^10+x^9-3*x^8-x^6-x^4-x^3+1)/

%p (x^14+6*x^13+2*x^11-6*x^10+x^9-3*x^8-x^6-x^4-x^3+1):

%p a:= n-> coeff(series(gf, x, n+1), x, n):

%p seq(a(n), n=0..66);

%Y Cf. A174249, A343529, A349187, A352421, A358933.

%K nonn,easy

%O 0,11

%A _Alois P. Heinz_, Apr 20 2023