A355588 -id:A355588 - cp's OEIS frontend

A355585 T(j,k) are the numerators s in the representation R = s/t + (2sqrt(3)/Pi)u/v of the resistance between two nodes separated by the distance (j,k) in an infinite triangular lattice of one-ohm resistors, where T(j,k), j >= 0, 0 <= k <= floor(j/2) is an irregular triangle read by rows.

0, 1, 8, -2, 27, -5, 928, -70, 16, 11249, -2671, 123, 46872, -34354, 5992, -438, 1792225, -445535, 28075, -10303, 23152256, -5824226, 1168304, -178754, 38336, 100685835, -25547957, 5343755, -885717, 101355, 3970817992, -338056246, 72962904, -12914726, 1825464, -386166

Offset: 0

Hugo Pfoertner, Jul 09 2022

The distance vector (j,k) is defined in an oblique coordinate system with an angle of 120 degrees between the axes, see e.g. A307012.
Atkinson and Steenwijk (1999) (see links in A211074) provided a generalization of the method used to calculate the resistance between two arbitrary nodes in an infinite square lattice of one-ohm resistors to infinite triangular lattices. Similar to the square lattice, the integral describing the resistance distance between nodes can exactly be represented by an expression of the form given in the name of this sequence with integer coefficients. Atkinson and Steenwijk, page 489, provided results for j <= 3 found by evaluation of the integral (17) (given below) and application of Mathematica's "Simplify" function.
R(j,k) = (1/Pi) * Integral_{y=0..Pi/2} (1 - exp(-|j-k|*x)*cos((j+k)*y)) / (sinh(x)*cos(y)) dy, with x = arccosh(2/cos(y)-cos(y)).
It would be useful to know whether, since the publication cited, a recurrence analogous to that known for the square lattice (used in A355565) for determining the coefficients has also been found for the triangular lattice.
The results in this sequence were found by systematic parameter variation of u and v and continued fraction expansion of the difference from the exact value of the integral for the resistance distance to determine s/t.

			The triangle begins:
          0;
          1;
          8,        -2;
         27,        -5;
        928,       -70,      16;
      11249,     -2671,     123;
      46872,    -34354,    5992,    -438;
    1792225,   -445535,   28075,  -10303;
   23152256,  -5824226, 1168304, -178754,  38336;
  100685835, -25547957, 5343755, -885717, 101355;
. The combined triangles used to calculate the resistances are:
   \ j                0              |                 1               |
   k\---------- s/t ----------- u/v -|----------- s/t ----------- u/v -|
   0|           0/1             0/ 1 |             .               .   |
   1|           1/3             0/ 1 |             .               .   |
   2|           8/3            -2/ 1 |           -2/3             1/ 1 |
   3|          27/1           -24/ 1 |           -5/1             5/ 1 |
   4|         928/3          -280/ 1 |          -70/1            64/ 1 |
   5|       11249/3         -3400/ 1 |        -2671/3           808/ 1 |
   6|       46872/1       -212538/ 5 |       -34354/3         51929/ 5 |
   7|     1792225/3      -2708944/ 5 |      -445535/3        673429/ 5 |
   8|    23152256/3    -244962336/35 |     -5824226/3      61623224/35 |
   9|   100685835/1   -3195918288/35 |    -25547957/1     810930216/35 |
  10|  3970817992/3  -42013225014/35 |   -338056246/1    2146081719/ 7 |
  11| 52514317745/3 -111125508824/ 7 | -13481564911/3  142641647567/35 |
.
continued
   \ j             2              |               3            |
   k\-------- s/t ---------- u/v -|--------- s/t -------- u/v -|
   4|        16/1          -14/ 1 |           .            .   |
   5|       123/1         -111/ 1 |           .            .   |
   6|      5992/3        -9054/ 5 |       -438/1       1989/5  |
   7|     28075/1      -127303/ 5 |     -10303/3      15576/5  |
   8|   1168304/3    -12361214/35 |    -178754/3    1891328/35 |
   9|   5343755/1   -169618717/35 |    -885717/1   28113999/35 |
  10|  72962904/1  -2315951182/35 |  -12914726/1   81986531/ 7 |
  11| 993810715/1 -31545031729/35 | -184858117/1 5867671888/35 |
.
continued
   \ j           4             |             5           |
   k\------- s/t -------- u/v -|------- s/t ------- u/v -|
   8|    38336/3    -405592/35 |         .           .   |
   9|   101355/1   -3217136/35 |         .           .   |
  10|  1825464/1  -57942922/35 |  -386166/1  12257507/35 |
  11| 28123355/1 -892677136/35 | -3085317/1  97932579/35 |
.
Using the terms for (j,k) = (10,5) with {s, t, u, v} = {-386166, 1, 12257507, 35} the resistance is R = T(10,5)/A355586(10,5) + (2*sqrt(3)/Pi) * A355587(10,5)/A355588(10,5) = -386166/1 + (2*sqrt(3)/Pi)*12257507/35 = 0.731139136228538824636... . This equals the integral for the resistance distance R(j,k) after substitution of j=10 and k=5.

See A211074 for more references and links (with alternatives).

D. Atkinson and F. J. van Steenwijk, Infinite resistive lattices, Am. J. Phys. 67 (1999), 486-492.
R. J. Mathar, Recurrence for the Atkinson-Steenwijk Integrals for Resistors in the Infinite Triangular Lattice, viXra:2208.0111 (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].

A355586 are the corresponding denominators t.
A355587 and A355588 are u and v.
Cf. A307012 (discussion of oblique coordinate system).
Cf. A084768 (when divided by 3 apparently gives the difference between successive values of s/t in column 0).
Cf. A355565, A355566, A355567 (similar problem for the square lattice).

Rtri(n,p)={my(alphat(beta)=acosh(2/cos(beta)-cos(beta))); intnum (beta=0, Pi/2, (1 - exp (-abs(n-p) * alphat(beta))*cos((n+p)*beta)) / (cos(beta)*sinh(alphat(beta)))) / Pi};
searchr (target, maxn=1000000, maxd=10, maxrat=1000, minn=0, mind=1) = {my (Rcons=2*sqrt(3)/Pi, delta=oo); for (d=mind, maxd, my(PP=Rcons/d); for (nn=minn, maxn, foreach ([-nn,nn], n, my (P=PP*n, T=target-P, Q = bestappr(T,maxrat), D=abs(target-P-Q)); if(D

\\ Alternative method using a recurrence; calculates triangle of s/t
jk(j,k) = {my(jj=j,kk=k); if(k<1,jj=j-k+1;kk=2-k); my(km=(jj+1)/2); if(kk>km, kk=2*km-kk); [jj,kk]};
D(n) = subst(pollegendre(n), 'x, 7);
ST(nend) = {my(nmax=nend+1, N=matrix(nmax,(nmax+1)\2)); for (n=2, nmax, N[n,1]=(1/3) * sum(k=0,n-2,D(k))); for (n=3, nmax, N[n,2] = (1/2)*(6*N[n-1,1] - 2*N[jk(n-1,2)[1],jk(n-1,2)[2]] - N[n-2,1] - N[n,1])); for (n=5, nmax, for (m=3, (n+1)\2, N[n,m] = 6*N[jk(n-1,m-1)[1],jk(n-1,m-1)[2]] - N[jk(n-1,m)[1],jk(n-1,m)[2]] - N[jk(n-2,m-1)[1],jk(n-2,m-1)[2]] - N[jk(n-2,m-2)[1],jk(n-2,m-2)[2]] - N[jk(n-1,m-2)[1],jk(n-1,m-2)[2]] - N[jk(n,m-1)[1],jk(n,m-1)[2]] )); N};
ST(11)

T(n,0)/A355586(n,0) = T(n-1,0)/A355586(n-1,0) + A084768(n-1)/3 for n>=1 (conjectured).

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.

Original entry on oeis.org

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

A355586 T(j,k) are the denominators t in the representation R = s/t + (2sqrt(3)/Pi)u/v of the resistance between two nodes separated by the distance (j,k) in an infinite triangular lattice of one-ohm resistors, where T(j,k), j >= 0, 0 <= k <= floor(j/2) is an irregular triangle read by rows.

Original entry on oeis.org

1, 3, 3, 3, 1, 1, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 3, 3, 1, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 3, 1, 3, 1, 1, 1, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 1, 3, 3, 1, 1, 3, 3, 3, 3, 1

Offset: 0

Hugo Pfoertner, Jul 09 2022

See A355585 for more information.

			The triangle begins:
  1;
  3;
  3, 3;
  1, 1;
  3, 1, 1;
  3, 3, 1;
  1, 3, 3, 1;
  3, 3, 1, 3;
  3, 3, 3, 3, 3;
  1, 1, 1, 1, 1;

See A211074 for references and links.

A355585 are the corresponding numerators.

A355587 and A355588 are u and v.

PARI
```
See A355585.
```

A355587 T(j,k) are the numerators u in the representation R = s/t + (2sqrt(3)/Pi)u/v of the resistance between two nodes separated by the distance (j,k) in an infinite triangular lattice of one-ohm resistors, where T(j,k), j >= 0, 0 <= k <= floor(j/2) is an irregular triangle read by rows.

Original entry on oeis.org

0, 0, -2, 1, -24, 5, -280, 64, -14, -3400, 808, -111, -212538, 51929, -9054, 1989, -2708944, 673429, -127303, 15576, -244962336, 61623224, -12361214, 1891328, -405592, -3195918288, 810930216, -169618717, 28113999, -3217136, -42013225014, 2146081719, -2315951182, 81986531, -57942922, 12257507

Offset: 0

Hugo Pfoertner, Jul 09 2022

See A355585 for more information.

			The triangle begins:
            0;
            0;
           -2,         1;
          -24,         5;
         -280,        64,        -14;
        -3400,       808,       -111;
      -212538,     51929,      -9054,     1989;
     -2708944,    673429,    -127303,    15576;
   -244962336,  61623224,  -12361214,  1891328,  -405592;
  -3195918288, 810930216, -169618717, 28113999, -3217136;

R. J. Mathar, Recurrence for the Atkinson-Steenwijk Integrals for Resistors in the Infinite Triangular Lattice, viXra:2208.0111 (2022).

A355588 are the corresponding denominators v.

A355585 and A355586 are s and t.

Rtri(n, p) = {my(alphat(beta)=acosh(2/cos(beta)-cos(beta))); intnum (beta=0, Pi/2, (1 - exp (-abs(n-p) * alphat(beta))*cos((n+p)*beta)) / (cos(beta)*sinh(alphat(beta)))) / Pi};
jk(j,k) = {my(jj=j,kk=k); if(k<1, jj=j-k+1; kk=2-k); my(km=(jj+1)/2); if(kk>km, kk=2*km-kk); [jj,kk]};
D(n) = subst(pollegendre(n), 'x, 7);
uv(k) = (Rtri(k,0) - sum(j=0, k-1, D(j))/3) / (2*sqrt(3)/Pi);
poddpri(primax) = {my(pp=1,p=2); while (p<=primax, p=nextprime(p+1); pp*=p); pp};
UV(nend) = { my(nmax=nend+1,M=matrix(nmax,(nmax+1)\2)); for (n=3, nmax, M[n,1] = bestappr(uv(n-1),poddpri(n-1))); for (n=3, nmax, M[n,2]=(1/2)*(6*M[n-1,1] - 2*M[jk(n-1,2)[1],jk(n-1,2)[2]] - M[n-2,1] - M[n,1])); for (n=5, nmax, for (m=3,(n+1)\2, M[n,m] = 6*M[jk(n-1,m-1)[1],jk(n-1,m-1)[2]] - M[jk(n-1,m)[1],jk(n-1,m)[2]] - M[jk(n-2,m-1)[1],jk(n-2,m-1)[2]] - M[jk(n-2,m-2)[1],jk(n-2,m-2)[2]] - M[jk(n-1,m-2)[1],jk(n-1,m-2)[2]] - M[jk(n,m-1)[1],jk(n,m-1)[2]] )); M};
UV(11)

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];

A356203 a(n) is the first component x of the distance vector (x,y) in an oblique 120-degree coordinate system, 0 <= y <= x, between two nodes of an infinite triangular 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 A356204(n).

Original entry on oeis.org

0, 43, 9615, 2299822, 507491696, 118805048562, 25315296119626, 5959615271620724

Offset: 0

Hugo Pfoertner, Aug 13 2022

			  n                   x                  y    R(x,y) - n
  0                   0                  0    0
  1                  43                 18    5.033*10^(-6)
  2                9615               2536    1.848*10^(-10)
  3             2299822            1136101   -3.120*10^(-14)
  4           507491696          119227930    5.751*10^(-19)
  5        118805048562        33636581266    5.618*10^(-23)
  6      25315296119626      1774960492720    8.406*10^(-29)
  7    5959615271620724    685318499093455    2.526*10^(-32)

Cf. A355585, A355586, A355587, A355588, A355589, A355954.

Cf. A356201, A356202 (similar for square lattice).

A356204 a(n) is the second component y of the distance vector (x,y) in an oblique 120-degree coordinate system, 0 <= y <= x, between two nodes of an infinite triangular 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 A356203(n).

Original entry on oeis.org

0, 18, 2536, 1136101, 119227930, 33636581266, 1774960492720, 685318499093455

Offset: 0

Hugo Pfoertner, Aug 13 2022

			See table in A356203.

Cf. A355585, A355586, A355587, A355588, A355589, A355954.

Cf. A356201, A356202 (similar for square lattice).

A355951 Negated column 0 of the irregular triangle A355587.

Original entry on oeis.org

0, 0, 2, 24, 280, 3400, 212538, 2708944, 244962336, 3195918288, 42013225014, 111125508824, 11603576403816, 30966112647080, 188641282541015866, 2532986569522773024, 34096877865475065728, 459984329860282638816, 105694712757690117569946, 1431044069320995796765272, 73738714208458783084303688

Offset: 0

Hugo Pfoertner, Jul 21 2022

a(n) are the numerators u in the representation R = s/t - (2*sqrt(3)/Pi)*u/v of the resistance between two nodes with distance n on the same grid line in an infinite triangular lattice of one-ohm resistors. The corresponding denominators are A355952. s(n)/t(n) = (1/3)*Sum_{k=0..n-1} A084768(k-1) for n >= 0.

R(n) > 1 [ohm] for n >= 38. Cserti (2000, page 11) claims that R(n) is logarithmically divergent for large values of n.

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.

Cf. A355587, A355952 (denominators).

Cf. A084768, A355585, A355586, A355588.

Rtri(n, p) = {my(alphat(beta)=acosh(2/cos(beta)-cos(beta))); intnum (beta=0, Pi/2, (1 - exp (-abs(n-p) * alphat(beta))*cos((n+p)*beta)) / (cos(beta)*sinh(alphat(beta)))) / Pi};
D(n) = subst(pollegendre(n), x, 7);
uv(k) = (Rtri(k) - sum(j=0, k-1, D(j))/3) / (2*sqrt(3)/Pi);
poddpri(primax) = {my(pp=1,p=2); while (p<=primax, p=nextprime(p+1); pp*=p); pp};
for (k=0, 20, print1(-numerator(bestappr(uv(k),poddpri(k))), ", "))
\\ for A355952 replace by
\\ for (k=0, 20, print1(denominator(bestappr(uv(k),poddpri(k))),", "))

cp's OEIS Frontend

A355952 Column 0 of the irregular triangle A355588.

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A355585 T(j,k) are the numerators s in the representation R = s/t + (2*sqrt(3)/Pi)*u/v of the resistance between two nodes separated by the distance (j,k) in an infinite triangular lattice of one-ohm resistors, where T(j,k), j >= 0, 0 <= k <= floor(j/2) is an irregular triangle read by rows.

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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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A355586 T(j,k) are the denominators t in the representation R = s/t + (2*sqrt(3)/Pi)*u/v of the resistance between two nodes separated by the distance (j,k) in an infinite triangular lattice of one-ohm resistors, where T(j,k), j >= 0, 0 <= k <= floor(j/2) is an irregular triangle read by rows.

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A355587 T(j,k) are the numerators u in the representation R = s/t + (2*sqrt(3)/Pi)*u/v of the resistance between two nodes separated by the distance (j,k) in an infinite triangular lattice of one-ohm resistors, where T(j,k), j >= 0, 0 <= k <= floor(j/2) is an irregular triangle read by rows.

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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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A355951 Negated column 0 of the irregular triangle A355587.

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A355585 T(j,k) are the numerators s in the representation R = s/t + (2sqrt(3)/Pi)u/v of the resistance between two nodes separated by the distance (j,k) in an infinite triangular lattice of one-ohm resistors, where T(j,k), j >= 0, 0 <= k <= floor(j/2) is an irregular triangle read by rows.

A355586 T(j,k) are the denominators t in the representation R = s/t + (2sqrt(3)/Pi)u/v of the resistance between two nodes separated by the distance (j,k) in an infinite triangular lattice of one-ohm resistors, where T(j,k), j >= 0, 0 <= k <= floor(j/2) is an irregular triangle read by rows.

A355587 T(j,k) are the numerators u in the representation R = s/t + (2sqrt(3)/Pi)u/v of the resistance between two nodes separated by the distance (j,k) in an infinite triangular lattice of one-ohm resistors, where T(j,k), j >= 0, 0 <= k <= floor(j/2) is an irregular triangle read by rows.