A355565 T(j,k) are the numerators s in the representation R = s/t + (2/Pi)*u/v of the resistance between two nodes separated by the distance vector (j,k) in an infinite square lattice of one-ohm resistors, where T(j,k), j >= 0, 0 <= k <= j, is a triangle read by rows.
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, 11073, -4376, 4321, -660, 241, -12, 1, 0, 29856, -48833, 13342, -9681, 1144, -337, 14, -1, 0, 325441, -136488, 160929, -33188, 18929, -1820, 449, -16, 1, 0
Offset: 0
Examples
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.
References
- See A211074 for more references and links.
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
Crossrefs
Programs
-
Maple
See link.
-
Mathematica
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 (p0, 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())
Formula
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
Comments