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.
Similar content being viewed by others
References
Amengual A, The intriguing properties of the equivalent resistances of n equal resistors combined in series and in parallel, Am. J. Phys. 68(2) (2000) 175–179
Asad J H, Hijjawi R S, Sakaj A and Khalifeh J M, Resistance calculation for an infinite simple cubic lattice application of Green’s function, Int. J. Theor. Phys. 43(11) (2004) 2223–2235
Asad J H, Hijjawi R S, Sakaj A and Khalifeh J M, Remarks on perturbation of infinite networks of identical resistors, Int. J. Theor. Phys. 44(4) (2005) 471–483
Boylestad R L, Introductory Circuit Analysis, 11th ed. (Pearson International, Prentice Hall) (2007)
Cristian C and Zaharescu A, The Haros-Farey sequence at two hundred years, Acta Universitatis Apulensis Math. Inform. 5 (2003) 1–38
Dunlap R A, The Golden Ratio and Fibonacci Numbers (World Scientific) (1997)
Hardy G H and Wright E M, An Introduction to the Theory of Numbers (London: Oxford University Press) (2008)
Khan S A, Lecture Notes in Physics, Salalah College of Technology E-Learning Website. Available at http://www.sct.edu.om/ (2010)
March R H, Polygons of resistors and convergent series, Am. J. Phys. 61(10) (1993) 900–901
Niven I, Zuckerman H S and Montgomery H L, An Introduction to the Theory of Numbers 5th ed. (New York: John Wiley) (2000)
Sloane N J A (ed.), The On-Line Encyclopedia of Integer Sequences (2008), available at http://www.research.att.com/~njas/sequences/ and http://oeis.org/; The OEIS Foundation Inc. http://www.oeisf.org/ (2010)
Srinivasan T P, Fibonacci sequence, golden ratio, and a network of resistors, Am. J. Phys. 60(5) (1992) 461–462
Van Steenwijk F J, Equivalent resistors of polyhedral resistive structures, Am. J. Phys. 66(1) (1998) 90–91
Weisstein E W, CRC Concise Encyclopedia of Mathematics, p. 1040; P. Chandra and E. W. Weisstein, Fibonacci Number, from MathWorld–A Wolfram Web Resource. Available at http://mathworld.wolfram.com/FibonacciNumber.html
Sequence A000045: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, ..., Sloane N J A, Fibonacci Numbers, Sequence A000045 in N. J. A. Sloane’s The On-Line Encyclopedia of Integer Sequences (2008), available at http://oeis.org/A000045
Sequence A000084: 1, 2, 4, 10, 24, 66, 180, 522, 1532, 4624, 14136, 43930, 137908, 437502, 1399068, 4507352, 14611576, 47633486, 156047204, 513477502, 1696305728, 5623993944, ..., Sloane N J A, Number of series-parallel networks with n unlabeled edges; also called yoke-chains by Cayley and MacMahon, Sequence A000084 in N. J. A. Sloane’s The On-Line Encyclopedia of Integer Sequences (2008), available at http://oeis.org/A000084
Sequence A005728: 2, 3, 5, 7, 11, 13, 19, 23, 29, 33, 43, 47, 59, 65, 73, 81, 97, 103, 121, 129, 141, 151, 173, 181, 201, 213, 231, 243, 271, 279, 309, 325, 345, 361, 385, 397, 433, 451, 475, 491, 531, 543, 585, 605, 629, 651, 697, 713, ..., Sloane N J A, Haros-Farey Sequence, Sequence A005728 in N. J. A. Sloane’s The On-Line Encyclopedia of Integer Sequences (2008), available at http://oeis.org/A005728
Sequence A048211: 1, 2, 4, 9, 22, 53, 131, 337, 869, 2213, 5691, 14517, 37017, 93731, 237465, 601093, 1519815, 3842575, 9720769, 24599577, 62283535, 157807915, 400094029, ..., Bartoletti T (more terms by Layman J W and Schoenfield J E), The number of distinct resistances that can be produced from a circuit of n equal resistors, Sequence A048211 in N. J. A. Sloane’s The On-Line Encyclopedia of Integer Sequences (2008), available at http://oeis.org/A048211
Sequence A153588: 1, 3, 7, 15, 35, 77, 179, 429, 1039, 2525, 6235, 15463, 38513, 96231, 241519, 607339, ..., Number of resistance values that can be constructed using n 1-ohm resistances by arranging them in an arbitrary series-parallel arrangement, Sequence A153588 in N. J. A. Sloane’s The On-Line Encyclopedia of Integer Sequences (2008), available at http://oeis.org/A153588
Sequence A174283: 1, 2, 4, 9, 23, 57, 151, 409, ..., Khan S A, Order of the set of distinct resistances that can be produced using n equal resistors in, series, parallel and/or bridge configurations, Sequence A174283 in N. J. A. Sloane’s The On-Line Encyclopedia of Integer Sequences (2008), available at http://oeis.org/A174283
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
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
Sequence A176498: 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 9, 24, 58, 124, 312, ..., Khan S A, Number of elements less than half in 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 A176498 in N. J. A. Sloane’s The On-Line Encyclopedia of Integer Sequences (2008), available at http://oeis.org/A176498
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
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
KHAN, S.A. Farey sequences and resistor networks. Proc Math Sci 122, 153–162 (2012). https://doi.org/10.1007/s12044-012-0066-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12044-012-0066-7