0,4

The recurrence given by Cserti (2000), page 5, (32) is used to calculate the resistance between two arbitrarily spaced nodes in an infinite square lattice whose edges are replaced by one-ohm resistors. The lower triangle, including the diagonal, in Table I of Atkinson and Steenwijk (1999), page 487, is reproduced. The solution to the resistor grid problem shown in the xkcd Web Comic #356 "Nerd Sniping", provided in A211074, is the special case (j,k) = (2,1).

Using the terms of A280079 and A280317 as pairs of grid indices leads to strictly increasing resistances, i.e., R(A280079(m),A280317(m)) > R(A280079(i),A280317(i)) for m > i. This implies that for grid points on the same radius the resistance increases with the circumferential angle between 0 and Pi/4. The further dependence of the resistance along the circumferential angle with a fixed radius results from symmetry. - Hugo Pfoertner, Aug 31 2022

See A211074 for more references and links.

Rainer Rosenthal, Table of n, a(n) for n = 0..135, rows 0..15 of triangle, flattened.

J. Cserti, Application of the lattice Green's function for calculating the resistance of infinite networks of resistors, arXiv:cond-mat/9909120 [cond-mat.mes-hall], 1999-2000.

Hugo Pfoertner, Grid points sorted by increasing R values, (2022).

Hugo Pfoertner, PARI program for inverse problem, (2022). Finds the grid point [x,y] that leads to the best approximation of a given resistance distance R (ohms) between [0,0] and [x,y].

Physics Stack Exchange, On this infinite grid of resistors, what's the equivalent resistance? Answer by user PBS, Apr 21 2018.

Rainer Rosenthal, Maple program

The resistance for the distance vector (j,k) is R(j,k) = T(j,k)/(1+mod(j+k,2)) +(2/Pi)*A355566(j,k)/A355567(j,k), avoiding the use of A131406.

From Rainer Rosenthal, Aug 04 2022: (Start)

R(0,0) = 0; R(1,0) = 1/2.

R(n,n) = R(n-1,n-1) + (2/Pi)/(2*n-1) for n >= 1.

R(j,k) = R(k,j) and R(-j,k) = R(j,k).

4*R(j,k) = R(j-1,k) + R(j+1,k) + R(j,k-1) + R(j,k+1) for (j,k) != (0,0).

(End)

T(j+1,0) = A089165(j)/(1 + mod(j,2)) for j >= 0. - Hugo Pfoertner, Aug 21 2022

The triangle begins:

0;

1, 0;

2, -1, 0;

17, -4, 1, 0;

40, -49, 6, -1, 0;

401, -140, 97, -8, 1, 0;

1042, -1569, 336, -161, 10, -1, 0

.

The combined triangles used to calculate the resistances are:

\ k 0 | 1 | 2 | 3 |

\ s/t u/v | s/t u/v | s/t u/v | s/t u/v |

j \---------------|-----------------|---------------|--------------|

0 | 0 0 | . . | . . | . . |

1 | 1/2 0 | 0 1 | . . | . . |

2 | 2 -2 | -1/2 2 | 0 4/3 | . . |

3 | 17/2 -12 | -4 23/3 | 1/2 2/3 | 0 23/15 |

4 | 40 -184/3 | - 49/2 40 | 6 -118/15 | -1/2 12/5 |

5 | 401/2 -940/3 | -140 3323/15 | 97/2 -1118/15 | -8 499/35 |

.

continued:

\ k 4 | 5 |

\ s/t u/v | s/t u/v |

j \-------------|--------------|

0 | . . | . . |

1 | . . | . . |

2 | . . | . . |

3 | . . | . . |

4 | 0 176/105 | . . |

5 | 1/2 20/21 | 0 563/315 |

.

E.g., the resistance for a node distance vector (4,1) is R = T(4,1)/A131406(5,2) + (2/Pi)*A355566(4,1)/A355567(4,1) = -49/2 + (2/Pi)*40/1 = 80/Pi - 49/2.

See link.

alphas[beta_] :=

Log[2 - Cos[beta] + Sqrt[3 + Cos[beta]*(Cos[beta] - 4)]];

Rsqu[n_, p_] :=

Simplify[(1/Pi)*

Integrate[(1 - Exp[-Abs[n]*alphas[beta]]*Cos[p*beta])/

Sinh[alphas[beta]], {beta, 0, Pi}]];

Table[Rsqu[n, k], {n, 0, 4}, {k, 0, n}] // TableForm (* Hugo Pfoertner, Aug 21 2022, calculates R, after Atkinson and Steenwijk *)

(PARI) R(m, p, x=pi) = {if (m==0 && p==0, return(0)); if (m==1 && p==0, return(1/2)); if (m==1 && p==1, return(2/x)); if(m==p, my(mm=m-1); return(R(mm, mm)*4*mm/(2*mm+1) - R(mm-1, mm-1)*(2*mm-1)/(2*mm+1))); if (p==(m-1), my(mm=m-1); return(2*R(mm, mm) - R(mm, mm-1))); if (p==0, my(mm=m-1); return(4*R(mm, 0) - R(mm-1, 0) - 2*R(mm, 1))); if (p<m && p>0, my(mm=m-1); return(4*R(mm, p) - R(mm-1, p) - R(mm, p+1) - R(mm, p-1)))};

for(j=0, 9, for(k=0, j, my(q=pi*R(j, k, pi)); print1(numerator(polcoef(q, 1, pi)), ", ")); print())

A131406 are the corresponding denominators t, with indices shifted by 1.

A355566 and A355567 are u and v.

Cf. A025547, A025550, A089165, A211074, A355953, A355955, A356201, A356202.

Cf. A355585, A355586, A355587, A355588 (same problem for the infinite triangular lattice).

Cf. A280079, A280317.

Sequence in context: A129065 A307804 A361718 * A202700 A024026 A380252

Adjacent sequences: A355562 A355563 A355564 * A355566 A355567 A355568

tabl,frac,sign

Hugo Pfoertner, Jul 07 2022

approved