A340726 Maximum power V_s*A_s consumed by an electrical network with n unit resistors and input voltage V_s and current A_s constrained to be exact integers which are coprime, and such that all currents between nodes are integers.

1, 2, 6, 15, 42, 143, 399, 1190, 4209, 13130, 41591, 118590, 404471, 1158696, 3893831, 12222320, 39428991, 123471920, 397952081, 1297210320

Offset: 1

Author

Rainer Rosenthal, Jan 17 2021

Keywords

• nonn
• hard
• more
• nice

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).

Examples

			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
		

Links

• Rainer Rosenthal, From_Quilt_to_Net
• Squaring.Net 2020, Stuart Anderson, Squared Rectangle and Smith Diagram
• Index to sequences related to resistances.

Crossrefs

Cf. A180414, A337517, A338601, A338602, A338861.

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