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A048211
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Number of distinct resistances that can be produced from a circuit of n equal resistors.
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17
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1, 2, 4, 9, 22, 53, 131, 337, 869, 2213, 5691, 14517, 37017, 93731, 237465, 601093, 1519815, 3842575, 9720769, 24599577, 62283535, 157807915, 400094029
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Found by exhaustive search. Program produces all values that are combinations of two binary operators a() and b() (here "sum" and "reciprocal sum of reciprocals") over n occurrences of 1. E.g. given 4 occurrences of 1, the code forms all allowable postfix forms, such as 1 1 1 1 a a a and 1 1 b 1 1 a b, etc. Each resulting form is then evaluated according to the definitions for a and b.
Each resistance that can be constructed from n 1-ohm resistors in a circuit can be written as the ratio of two positive integers, neither of which exceeds the (n+1)st Fibonacci number. E.g., for n=4, the 9 resistances that can be constructed can be written as 1/4, 2/5, 3/5, 3/4, 1/1, 4/3, 5/3, 5/2, 4/1 using no numerator or denominator larger than Fib(n+1) = Fib(5) = 5. If a resistance x can be constructed from n 1-ohm resistors, then a resistance 1/x can also be constructed from n 1-ohm resistors. - Jon E. Schoenfield (jonscho(AT)hiwaay.net), Aug 06 2006
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REFERENCES
| Antoni Amengual, The intriguing properties of the equivalent resistances of n equal resistors combined in series and in parallel, American Journal of Physics, 68(2), 175-179 (February 2000). Digital Object Identifier (DOI): http://dx.doi.org/10.1119/1.19396 [From Sameen Ahmed KHAN (rohelakhan(AT)yahoo.com), Apr 27 2010]
Sameen Ahmed Khan, The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel, E-Print archive: http://arxiv.org/abs/1004.3346/ (21 April 2010). [From Sameen Ahmed KHAN (rohelakhan(AT)yahoo.com), Apr 27 2010]
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LINKS
| Sameen Ahmed KHAN, Mathematica program
Sameen Ahmed KHAN, Mathematica notebook for A048211 and A000084
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EXAMPLE
| a(2) = 2 since given two 1-ohm resistors, a series circuit yields 2 ohms, while a parallel circuit yields 1/2 ohms.
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CROSSREFS
| Let T(x, n) = 1 if x can be constructed with n 1-ohm resistors in a circuit, 0 otherwise. Then A048211 is t(n) = sum(T(x, n)) for all x (x is necessarily rational). Let H(x, n) = 1 if T(x, n) = 1 and T(x, k) = 0 for all k < n, 0 otherwise. Then A051389 is h(n) = sum(H(x, n)) for all x (x is necessarily rational).
A153588, A174283, A174284, A174285 and A174286, A176497, A176498, A176499, A176500, A176501, A176502 [From Sameen Ahmed KHAN (rohelakhan(AT)yahoo.com), Apr 27 2010]
Sequence in context: A129875 A055094 A055729 * A098719 A198520 A115324
Adjacent sequences: A048208 A048209 A048210 * A048212 A048213 A048214
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KEYWORD
| nonn,nice,more
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AUTHOR
| Tony Bartoletti (azb(AT)llnl.gov)
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EXTENSIONS
| More terms from John W. Layman (layman(AT)math.vt.edu), Apr 06 2002
a(16) through a(21) from Jon E. Schoenfield (jonscho(AT)hiwaay.net), Aug 06 2006
a(22) from Jon E. Schoenfield (jonscho(AT)hiwaay.net), Aug 28 2006
a(23) from Jon E. Schoenfield (jonscho(AT)hiwaay.net), Apr 18 2010
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