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Farey sequences and resistor networks

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Abstract

In this article, we employ the Farey sequence and Fibonacci numbers to establish strict upper and lower bounds for the order of the set of equivalent resistances for a circuit constructed from n equal resistors combined in series and in parallel. The method is applicable for networks involving bridge and non-planar circuits.

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  21. Sequence A174284: 1, 3, 7, 15, 35, 79, 193, 489, ..., Khan S A, Order of the set of distinct resistances that can be produced using at most n equal resistors in series, parallel and/or bridge configurations, Sequence A174284 in N. J. A. Sloane’s The On-Line Encyclopedia of Integer Sequences (2008), available at http://oeis.org/A174284

  22. Sequence A176497: 0, 0, 0, 1, 4, 9, 25, 75, 195, 475, 1265, 3135, 7983, 19697, 50003, 126163, 317629, 802945, 2035619, 5158039, 13084381, 33240845, 84478199, ..., Khan S A, Order of the cross set which is the subset of the set of distinct resistances that can be produced using n equal resistors in series and/or parallel, Sequence A176497 in N. J. A. Sloane’s The On-Line Encyclopedia of Integer Sequences (2008), available at http://oeis.org/A176497

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  24. Sequence A176502: 1, 3, 7, 17, 37, 99, 243, 633, 1673, 4425, 11515, 30471, 80055, 210157, 553253, 1454817, 3821369, 10040187, ..., Khan S A, \(2 \mathcal{F}_m(I) - 1\), where m = F n + 1 and I = [1/n, 1], Sequence A176502 in N. J. A. Sloane’s The On-Line Encyclopedia of Integer Sequences (2008), available at http://oeis.org/A176502

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Correspondence to SAMEEN AHMED KHAN.

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KHAN, S.A. Farey sequences and resistor networks. Proc Math Sci 122, 153–162 (2012). https://doi.org/10.1007/s12044-012-0066-7

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  • DOI: https://doi.org/10.1007/s12044-012-0066-7

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