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T(n,m) is the numerator of the resistance between two nodes located at the end of a side of length n of a rectangular electric network of n*m quadratic meshes in which all edges are replaced by one-ohm resistors, where T(n,m) is a square array read by descending antidiagonals.
3

%I #10 Sep 16 2022 15:38:45

%S 3,11,4,41,5,13,153,26,267,26,571,68,181,192,149,2131,89,10609,1506,

%T 1171,138,7953,466,25059,251,155927,246,375,29681,1220,3869723,13852,

%U 759435,77948,75255,668,110771,1597,1334085,781778,109897,1949020,982871,24995,3523

%N T(n,m) is the numerator of the resistance between two nodes located at the end of a side of length n of a rectangular electric network of n*m quadratic meshes in which all edges are replaced by one-ohm resistors, where T(n,m) is a square array read by descending antidiagonals.

%F Limit_{n->oo} T(1,n)/A357116(1,n) = sqrt(3) - 1.

%e The array of resistances starts:

%e 3/4, 11/15, 41/56, 153/209, ... A001835(n-1)/A001353(n-1)

%e 4/3, 5/4, 26/21, 68/55, ...

%e 13/7, 267/161, 181/112, 10609/6603, ...

%e 26/11, 192/95, 1506/781, 251/132, ...

%e 149/52, 1171/495, 155927/70616, 759435/352583, ...

%e 138/41, 246/91, 77948/31529, 1949020/817991, ...

%e .

%e T(1,3)/A357116(1,3) = 41/56:

%e . _____ _____ _____

%e O--|__1__|--O--|__1__|--O--|__1__|--O-----O

%e | | | | |

%e | | | | | | | | |

%e |1| |1| |1| |1| 41/56 ohms

%e |_| |_| |_| |_| |

%e | _____ | _____ | _____ | |

%e O--|__1__|--O--|__1__|--O--|__1__|--O-----O

%e .

%e T(3,1)/A357116(3,1) = 13/7 T(2,2)/A357116(2,2) = 5/4

%e . _____ _____ _____

%e O--|__1__|--O-----O O--|__1__|--O--|__1__|--O-----O

%e | | | | | | |

%e | | | | | | | | | | | |

%e |1| |1| | |1| |1| |1| |

%e |_| |_| | |_| |_| |_| |

%e | _____ | | | _____ | _____ | |

%e O--|__1__|--O | O--|__1__|--O--|__1__|--O 5/4 ohms

%e | | | | | | |

%e | | | | 13/7 | | | | | | |

%e |1| |1| ohms |1| |1| |1| |

%e |_| |_| | |_| |_| |_| |

%e | _____ | | | _____ | _____ | |

%e O--|__1__|--O | O--|__1__|--O--|__1__|--O-----O

%e | | |

%e | | | | |

%e |1| |1| |

%e |_| |_| |

%e | _____ | |

%e O--|__1__|--O-----O

%Y A357116 are the corresponding denominators.

%Y Cf. A357113, A357114.

%K nonn,frac,tabl

%O 1,1

%A _Hugo Pfoertner_, Sep 15 2022