A355565 -id:A355565 - 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).

A211074 Decimal expansion of 4/Pi - 1/2.

Original entry on oeis.org

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

Offset: 0

Charles R Greathouse IV, May 02 2012

Equivalent resistance between two nodes on an infinite rectangular lattice of ideal unit resistors, where the nodes are separated by two resistors along one axis and one resistor on the other.

			0.77323954473516268615107010698011489627567716592365158998133875247117...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Author?, Infinite 2D square grid of 1-ohm resistors, April 30 2004.
D. Atkinson and F. J. van Steenwijk, Infinite resistive lattices, Aug 14 1998.
D. Atkinson and F. J. van Steenwijk, Infinite resistive lattices, Am. J. Phys. 67 (1999), 486-492.
Matthew Beckler, Infinite Grid of Resistors - Solution by Simulation, Dec 16 2009.
Kevin Brown, Infinite Grid of Resistors.
J. Cserti, Application of the lattice Green's function for calculating the resistance of an infinite networks of resistors, arXiv:cond-mat/9909120 [cond-mat.mes-hall], 1999-2000.
Peter G. Doyle and J. Laurie Snell, Random walks and electric networks (2006), section 1.1.4.
Alan Eustace, Pencils down, people (official Google Blog article, archived on archive.org), Sep 30 2004.
Randall Munroe, Nerd Sniping, xkcd Web Comic #356, Dec 12 2007.
Michael Pusateri, Google Labs Aptitude Test, Sep 13 2004.
Index entries for transcendental numbers

Cf. A088538, A355565.

RealDigits[4/Pi - 1/2, 10, 87][[1]] (* modified by Harvey P. Dale, Jan 14 2015 *)

PARI
```
4/Pi-1/2
```

A355566 T(j,k) are the numerators u 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, 0, 1, -2, 2, 4, -12, 23, 2, 23, -184, 40, -118, 12, 176, -940, 3323, -1118, 499, 20, 563, -24526, 1234, -18412, 13462, -626, 118, 6508, -130424, 721937, -71230, 327143, -1312, 14369, 262, 88069, -4924064, 191776, -6601046, 2395676, -888568, 131972, -300766, 1624, 91072

Offset: 0

Hugo Pfoertner, Jul 07 2022

See A355565 for more information.

On the diagonal we have T(0,0) = 0 and T(n,n) = A350669(n-1) for n > 0. - Rainer Rosenthal, Aug 01 2022

			The triangle begins:
       0;
       0,    1;
      -2,    2,      4;
     -12,   23,      2,    23;
    -184,   40,   -118,    12,  176;
    -940, 3323,  -1118,   499,   20, 563;
  -24526, 1234, -18412, 13462, -626, 118, 6508;

See A211074 for references and links.

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

A355567 are the corresponding denominators v.
A355565 and A131406 (with changed offset) are s and t.
Cf. A350669.

\\ uses function R(m, p, x) given in A355565
for (j=0, 8, for (k=0, j, my(q=(pi/2)*R(j,k)); print1(numerator(polcoef(q,0,pi)),", ")); print())

A355567 T(j,k) are the denominators v 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

1, 1, 1, 1, 1, 3, 1, 3, 3, 15, 3, 1, 15, 5, 105, 3, 15, 15, 35, 21, 315, 15, 1, 35, 105, 45, 45, 3465, 15, 105, 21, 315, 7, 693, 231, 45045, 105, 5, 315, 315, 495, 495, 15015, 585, 45045, 7, 315, 45, 3465, 3465, 45045, 45045, 15015, 385, 765765, 315, 35, 3465, 495, 45045, 6435, 15015, 45045, 765765, 9945, 14549535

Offset: 0

Hugo Pfoertner, Jul 07 2022

See A355565 for more information.

On the diagonal we have T(0,0) = 1 and T(n,n) = A350670(n-1) for n > 0. - Rainer Rosenthal, Aug 01 2022

			The triangle begins:
   1;
   1,  1;
   1,  1,  3;
   1,  3,  3,  15;
   3,  1, 15,   5, 105;
   3, 15, 15,  35,  21, 315;
  15,  1, 35, 105,  45,  45, 3465

See A211074 for references and links.

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

A355566 are the corresponding numerators u.
A355565 and A131406 (with changed offset) are s and t.
Cf. A350670.

\\ uses function R(m, p, x) given in A355565
for (j=0, 8, for (k=0, j, my(q=(pi/2)*R(j, k)); print1(denominator(polcoef(q, 0, pi)), ", ")); print())

A355955 a(n) is the least distance of two nodes on the same grid line in an infinite square lattice of one-ohm resistors for which the resistance measured between the two nodes is greater than n ohms.

Original entry on oeis.org

1, 5, 107, 2460, 56922, 1317211, 30481165, 705355254, 16322409116

Offset: 0

Hugo Pfoertner, Jul 23 2022

The terms are obtained by a high-precision evaluation of the integral R(j,k) = (1/Pi) * Integral_{beta=0..Pi} (1 - exp(-abs(j)*alphas(beta))*cos(k*beta)) / sinh(alphas(beta)), with alphas(beta) = log(2 - cos(beta) + sqrt(3 + cos(beta)*(cos(beta) - 4))) such that floor(R(m-1,0)) < floor(R(m,0)). The values of m for which this condition is satisfied are the terms of the sequence. See Atkinson and van Steenwijk (1999, page 491, Appendix B) for a Mathematica implementation of the integral.

a(9) = 377711852375, found by solving R(x) - 9 = 0, using the asymptotic formula provided by Cserti (2000, page 5), R(x) = (log(x) + gamma + log(8)/2)/Pi, needs independent confirmation. gamma is A001620.

			a(0) = 1: R(1,0) = 1/2 is the first resistance > 0;
a(1) = 5: R(4,0) = 0.953987..., R(5,0) = 1.025804658...;
a(2) = 107: R(106,0) = 1.999103258858..., R(107,0) = 2.002092149977722...;
a(3) = 2460: R(2459,0) = 2.999894481..., R(2460,0) = 3.0000239019301...;
a(4) = 56922: R(56921,0) = 3.99999536602..., R(56922,0) =  4.0000009581... .

D. Atkinson and F. J. van Steenwijk, Infinite resistive lattices, Am. J. Phys. 67 (1999), 486-492. (See A211074 for an alternative link.)
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. A355565, A355589 (same problem for triangular lattice).

\\ can be used to calculate estimates of terms for n >= 2, using the asymptotic formula. For n <= 8 results identical to those using the exact evaluation of the full integral are produced, but equality for higher terms might not hold, although with extremely remote probability.
a355955_asymp(upto) = {my(c=2.2, Rsqasy(L)=(1/Pi)*(log(L)+Euler+log(8)/2), d, m); for (n=2, upto, d=exp(c*n); d=solve(x=0.5*d, 2.5*d, Rsqasy(x)-n); print1(ceil(d),", "); c=log(d)/n)};
a355955_asymp(8)

A355953 Decimal expansion of (gamma + log(8)/2)/Pi.

Original entry on oeis.org

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

Offset: 0

Hugo Pfoertner, Jul 26 2022

This constant is the additive part A in the asymptotic behavior of the resistance R between two nodes in an infinite square lattice of one-ohm resistors separated by the distance vector (i,j): R(i,j) = log(sqrt(i^2+j^2))/Pi + A. 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 1/2 ohms between 2 neighboring nodes as the main part.

See, e.g., Cserti (1999) formula (33) on page 5 and Appendix B, pages 15 and 16, for a derivation of the parts of the constant.

			0.5146868528272853708539691163207527193...

József Cserti, Application of the lattice Green's function for calculating the resistance of an infinite network of resistors, American Journal of Physics, Vol. 68, No. 10 (2000), pp. 896-906; arXiv preprint, arXiv:cond-mat/9909120 [cond-mat.mes-hall], 1999-2000.

Cf. A001620, A016631, A355955, A355954 (similar for triangular lattice).

Cf. A355565, A355566, A355567 (exact solutions for small distances).

RealDigits[(EulerGamma + Log[8]/2)/Pi, 10, 120][[1]] (* Amiram Eldar, Jun 18 2023 *)

PARI
```
(Euler + log(8)/2)/Pi
```

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

A357021 First coordinate x of points in the triangular lattice, sorted first by the distance from the origin and then by the circumferential angle phi restricted to the sector 0 <= phi < Pi/6. y is given in A357022.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 7, 8, 8, 8, 9, 9, 8, 9, 9, 10, 10, 10, 9, 10, 10, 11, 11, 11, 10, 11, 12, 12, 11, 12, 12, 11, 12, 13, 13, 12, 13, 13, 12, 13, 14, 14, 14, 14, 13, 14, 13, 15, 15, 14, 15, 15, 14, 15, 16, 16, 14, 16, 15

Offset: 1

Hugo Pfoertner, Sep 10 2022

nonn

The coordinates (x,y) are defined in an oblique coordinate system with an angle of 120 degrees between the axes, see e.g. A307012.
The distance from the origin is given by r = sqrt(x^2 - x*y + y^2), and the circumferential angle is phi = atan(sqrt(3)*y/(2*x - y)).
Using the pairs of terms of this sequence and of A357022(n) as grid indices in an infinite triangular lattice of one-ohm resistors leads to strictly increasing resistances against (0,0) (see A355585). This is similar to the role of A280079 and A280317 used as grid indices in the square lattice (see A355565).

			R is the resistance between a grid point (x,y) and (0,0) in an infinite triangular lattice of one-ohm resistors.
.
   n  x y  r^2   phi      R
              (degrees) (ohms)
   1  0 0   0          0.0000000000
   2  1 0   1    0.000 0.3333333333
   3  2 1   3   30.000 0.4359911242
   4  2 0   4    0.000 0.4613510850
   5  3 1   7   19.107 0.5132889542
   6  3 0   9    0.000 0.5362130198
   7  4 2  12   30.000 0.5627909282
   8  4 1  13   13.898 0.5700986140
   9  4 0  16    0.000 0.5891518971
  ...
  19  7 1  43    7.589 0.6800193341
  20  8 4  48   30.000 0.6901322715
  21  7 0  49    0.000 0.6920215369
  22  8 3  49   21.787 0.6920259223
  23  8 2  52   13.898 0.6974842443

Cf. A003136, A280079, A280317, A305575, A305576, A355565, A355585.

cp's OEIS Frontend

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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A211074 Decimal expansion of 4/Pi - 1/2.

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A355566 T(j,k) are the numerators u 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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A355567 T(j,k) are the denominators v 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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A355955 a(n) is the least distance of two nodes on the same grid line in an infinite square lattice of one-ohm resistors for which the resistance measured between the two nodes is greater than n ohms.

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A355953 Decimal expansion of (gamma + log(8)/2)/Pi.

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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).

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.