OFFSET
1,2
COMMENTS
This sequence is an analog of A338861. Equality a(n) = A338861(n) holds for small n only, see example.
Let V_s denote the specific voltage, i.e., the lowest integer voltage, which induces integer currents everywhere in the network. Denote by A_s the specific current, i.e., the corresponding total current.
A planar network with n unit resistors corresponds to a squared rectangle with height V_s and width A_s. The electrical power V_s*A_s therefore equals the area of that rectangle. In the historical overview (Stuart Anderson link) A_s is called complexity.
The corresponding rectangle tiling provides the optimal power rating of the 1 ohm resistors with respect to the specific voltage V_s and current A_s. See the picture From_Quilt_to_Net in the link section, which also provides insight in the "mysterious" correspondence between rectangle tilings and electric networks. For non-planar nets the idea of rectangle tilings can be widened to 'Cartesian squarings'. A Cartesian squaring is the dissection of the product P X Q of two finite sets into 'squaresets', i.e., sets A X B with A subset of P and B subset of Q, and card(A) = card(B). - Rainer Rosenthal, Dec 14 2022
Take the set SetA337517(n) of resistances, counted by A337517. For each resistance R multiply numerator and denominator. Conjecture: a(n) is the maximum of all these products. The reason is that common factors of V_s and A_s are quite rare (see the beautiful exceptional example with 21 resistors).
LINKS
Rainer Rosenthal, From_Quilt_to_Net
Squaring.Net 2020, Stuart Anderson, Squared Rectangle and Smith Diagram
EXAMPLE
n = 3:
Networks with 3 unit resistors have A337517(3) = 4 resistance values: {1/3, 3, 3/2, 2/3}. The maximum product numerator X denominator is 6.
n = 6:
Networks with 6 unit resistors have A337517(6) = 57 resistance values, where 11/13 and 13/11 are the resistances with maximum product numerator X denominator.
+-----------+-------------+
A | | |
/ \ | | |
(1) / \ (2) | 6 X 6 | 7 X 7 |
/ \ | | |
/ (3) \ | | |
o---------o +---------+-+ |
\ // | +-+-----+-------+
\ (5)// | 5 X 5 | | |
(4) \ //(6) | | 4 X 4 | 4 X 4 |
\ // | | | |
Z +---------+-------+-------+
___________________________________________________________________
Network with 6 unit resistors Corresponding rectangle tiling
total resistance 11/13 giving with 6 squares giving
a(6) = 11 X 13 = 143 A338861(6) = 143
n = 10:
With n = 10, non-planarity comes in, yielding a(10) > A338861(10).
The "culprit" here is the network with resistance A338601(9)/A338602(9) = 130/101, giving a(10) = 13130 > A338861(10) = 10920.
n = 21:
The electrical network corresponding to the perfect squared square A014530 has specific voltage V_s equal to specific current A_s, namely V_s = A_s = 112. Its power V_s*A_s = 12544 is far below the maximum a(20) > a(10) > 13000, and a(n) is certainly monotonically increasing. - Rainer Rosenthal, Mar 28 2021
CROSSREFS
KEYWORD
nonn,hard,more,nice
AUTHOR
Rainer Rosenthal, Jan 17 2021
EXTENSIONS
a(13)-a(17) from Hugo Pfoertner, Feb 08 2021
Definition corrected by Rainer Rosenthal, Mar 28 2021
a(18) from Hugo Pfoertner, Apr 09 2021
a(19)-a(20) from Hugo Pfoertner, Apr 16 2021
STATUS
approved