A355954 -id:A355954 - cp's OEIS frontend

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

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
```

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).

cp's OEIS Frontend

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

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