A355953 -id:A355953 - cp's OEIS frontend

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

Hugo Pfoertner, Jul 07 2022

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

			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 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

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.

Maple
```
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 *)

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())

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

A356201 a(n) is the first component x of the distance vector (x,y), x >= y >= 0, between two nodes of an infinite square lattice of one-ohm resistors, such that the resistance R between the two nodes is as close as possible to n ohms, i.e., abs(R - n) is minimized. y is A356202(n).

Original entry on oeis.org

0, 4, 106, 2384, 51196, 958170, 24341911, 636875169, 14536767750, 285039411789, 6322647312660, 202105291334913

Offset: 0

Hugo Pfoertner, Aug 01 2022

If more than one solution exists, the one maximizing x and minimizing y is chosen.

			   n                x              y   R(x,y) - n
   0                0              0   0
   1                4              2  -8.076*10^(-3)
   2              106              8   7.349*10^(-6)
   3             2384            606   2.206*10^(-8)
   4            51196          24881  -7.426*10^(-11)
   5           958170         903855   7.396*10^(-16)
   6         24341911       18345919  -7.814*10^(-16)
   7        636875169      303176603  -3.017*10^(-19)
   8      14536767750     7423167971   5.874*10^(-21)
   9     285039411789   247828120179  -2.461*10^(-24)
  10    6322647312660  6034957650107  -1.048*10^(-26)
  11  202105291334913  7948827377158   1.795*10^(-29)

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].

Cf. A355565, A355566, A355567, A355953, A355955.

Cf. A356203, A356204 (similar for triangular lattice).

\\ using the function Rsqlatt(R) from the linked program
for (k=0, 11, print1(Rsqlatt(k)[1], ", ")) \\ Hugo Pfoertner, Sep 09 2022

a(9)-a(11) from Hugo Pfoertner, Aug 22 2022

A356202 a(n) is the second component y of the distance vector (x,y), x >= y >= 0, between two nodes of an infinite square lattice of one-ohm resistors, such that the resistance R between the two nodes is as close as possible to n ohms, i.e., abs(R - n) is minimized. x is A356201(n).

Original entry on oeis.org

0, 2, 8, 606, 24881, 903855, 18345919, 303176603, 7423167971, 247828120179, 6034957650107, 7948827377158

Offset: 0

Hugo Pfoertner, Aug 01 2022

If more than one solution exists, the one maximizing x and minimizing y is chosen.

			See table in A356201.

Cf. A355565, A355566, A355567, A355953, A355955.

\\ using the function Rsqlatt(R) from program file linked in A356201
for (k=0, 11, print1(Rsqlatt(k)[2], ", ")) \\ Hugo Pfoertner, Sep 09 2022

a(9)-a(11) from Hugo Pfoertner, Aug 22 2022

A355954 Decimal expansion of the constant A in the asymptotic behavior R(d) = log(d)/(Pi*sqrt(3)) + A of the resistance between two nodes separated by the Euclidean distance d in an infinite triangular lattice of one-ohm resistors.

Original entry on oeis.org

3, 3, 4, 4, 1, 2, 0, 3, 1, 3, 9, 2, 4, 1, 9, 8, 0, 2, 0, 4, 3, 9, 1, 3, 9, 1, 2, 2, 2, 1, 2, 2, 7, 0, 8, 8, 1, 5, 4, 5, 6, 5, 1

Offset: 0

Hugo Pfoertner, Jul 26 2022

From an engineering point of view, this constant summand can be regarded as a kind of near-field contribution, which contains the well-known resistance of 1/3 ohms between 2 neighboring nodes as the main part.

The asymptotic formula is analogous to that known for the square lattice. The constant was determined by comparison with the exact integral (see A355589) for the resistance, evaluated for very large distances d (maximum approx. 10^9, for larger arguments the computational effort is no longer manageable). At the moment (July 2022) no representation in closed form is known. A derivation similar to the method used to determine A355953 might be applicable.

			0.3344120313924198...

Cf. A355589, A355953 (similar for square lattice).

Cf. A355585, A355586, A355587, A355588 (exact solutions for small distances).

alphat[beta_] := ArcCosh[2/Cos[beta] - Cos[beta]];
Rtri[n_, p_] :=
  SetAccuracy[1/(Pi), 150]*
   NIntegrate[(1 -
       Exp[-Abs[n - p]*alphat[beta]]*Cos[(n + p)*beta])/(Cos[
        beta]*Sinh[alphat[beta]]), {beta, 0, Pi/2},
    WorkingPrecision -> 150];
Rtri[3*10^8, 0] - SetAccuracy[Log[3*10^8]/(Pi* Sqrt[3]), 150];

cp's OEIS Frontend

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.

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A356201 a(n) is the first component x of the distance vector (x,y), x >= y >= 0, between two nodes of an infinite square lattice of one-ohm resistors, such that the resistance R between the two nodes is as close as possible to n ohms, i.e., abs(R - n) is minimized. y is A356202(n).

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A356202 a(n) is the second component y of the distance vector (x,y), x >= y >= 0, between two nodes of an infinite square lattice of one-ohm resistors, such that the resistance R between the two nodes is as close as possible to n ohms, i.e., abs(R - n) is minimized. x is A356201(n).

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Author

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Comments

Examples

Crossrefs

Programs

PARI

Extensions

A355954 Decimal expansion of the constant A in the asymptotic behavior R(d) = log(d)/(Pi*sqrt(3)) + A of the resistance between two nodes separated by the Euclidean distance d in an infinite triangular lattice of one-ohm resistors.

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Author

Keywords

Comments

Examples

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Programs

Mathematica