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%I #56 Dec 15 2022 13:50:22
%S 1,2,6,15,42,143,399,1190,4209,13130,41591,118590,404471,1158696,
%T 3893831,12222320,39428991,123471920,397952081,1297210320
%N 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.
%C This sequence is an analog of A338861. Equality a(n) = A338861(n) holds for small n only, see example.
%C 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.
%C 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.
%C 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
%C 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).
%H Rainer Rosenthal, <a href="/A340726/a340726.jpg">From_Quilt_to_Net</a>
%H Squaring.Net 2020, Stuart Anderson, <a href="http://www.squaring.net/history_theory/brooks_smith_stone_tutte_II.html">Squared Rectangle and Smith Diagram</a>
%H <a href="/index/Res#resistances">Index to sequences related to resistances</a>.
%e n = 3:
%e 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.
%e n = 6:
%e 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.
%e +-----------+-------------+
%e A | | |
%e / \ | | |
%e (1) / \ (2) | 6 X 6 | 7 X 7 |
%e / \ | | |
%e / (3) \ | | |
%e o---------o +---------+-+ |
%e \ // | +-+-----+-------+
%e \ (5)// | 5 X 5 | | |
%e (4) \ //(6) | | 4 X 4 | 4 X 4 |
%e \ // | | | |
%e Z +---------+-------+-------+
%e ___________________________________________________________________
%e Network with 6 unit resistors Corresponding rectangle tiling
%e total resistance 11/13 giving with 6 squares giving
%e a(6) = 11 X 13 = 143 A338861(6) = 143
%e n = 10:
%e With n = 10, non-planarity comes in, yielding a(10) > A338861(10).
%e The "culprit" here is the network with resistance A338601(9)/A338602(9) = 130/101, giving a(10) = 13130 > A338861(10) = 10920.
%e n = 21:
%e 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
%Y Cf. A180414, A337517, A338601, A338602, A338861.
%K nonn,hard,more,nice
%O 1,2
%A _Rainer Rosenthal_, Jan 17 2021
%E a(13)-a(17) from _Hugo Pfoertner_, Feb 08 2021
%E Definition corrected by _Rainer Rosenthal_, Mar 28 2021
%E a(18) from _Hugo Pfoertner_, Apr 09 2021
%E a(19)-a(20) from _Hugo Pfoertner_, Apr 16 2021
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