cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Hugo Pfoertner, Jul 23 2022

Keywords

Comments

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.

Examples

			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... .

Links

Crossrefs

Cf. A355565, A355589 (same problem for triangular lattice).

Programs

  • PARI
    \\ 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)

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

Views

Author

Hugo Pfoertner, Jul 26 2022

Keywords

Comments

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.

Examples

			0.3344120313924198...

Crossrefs

Cf. A355589, A355953 (similar for square lattice).
Cf. A355585, A355586, A355587, A355588 (exact solutions for small distances).

Programs

  • Mathematica
    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

Views

Author

Hugo Pfoertner, Aug 13 2022

Keywords

Examples

			  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)

Crossrefs

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

Views

Author

Hugo Pfoertner, Aug 13 2022

Keywords

Examples

			See table in A356203.

Crossrefs

Cf. A355585, A355586, A355587, A355588, A355589, A355954.
Cf. A356201, A356202 (similar for square lattice).
Showing 1-4 of 4 results.

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